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SOME STRUCTURE THEOREMS

FOR TOPOLOGICAL MACHINES

By
EUGENE MICHAEL NORRIS

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1969

Page

Introduction 1

Chapter

1. Preliminaries 3

1.1 Definitions 3
1.2 Some Isomorphism Theorems 11
1.3 A Schreier Refinement Theorem 15
1.4 Some Lattices Associated with an Act 18

2. Relative Ideals in Acts 21

2.1 Minimal Relative Ideals 23
2.2 Maximal Proper Relative Ideals 27

3. Categoricalities 33

3.1 Categories 34
3.2 Products of Acts 40
3.3 Coproducts and Free Acts 43
3.4 Inverse Limits of Compact, Totally
Disconnected Acts- : 48
3.5 An Embedding Theorem 54

Appendix 64

Bibliography 67

INTRODUCTION

This dissertation is not an introduction to the theory

of acts, since it is not sufficiently comprehensive. How-

ever, its contents particularly chapters one and two -

are written in the spirit of a prolegomenon to that theory.

In these two chapters we give the definitions of

admissible pair, due to Bednarek and Wallace [4],* and of

left, right and two-sided ideals for acts. These last seem

to be new. An admissible pair is the analogue for acts of

the notion of congruence for semigroups, and the notion of

ideal is fashioned upon the eponymous semigroup definition.

In this spirit, we investigate quotients of acts by admis-

sible pairs and in particular by closed ideals to obtain

an analogue of the ideal theory for topological semigroups.

The Noetherian isomorphism theorems are seen to hold as a

consequence of an act-theoretic version of a theorem of

Sierpinski on the completion of certain mapping diagrams,

and a Schreier refinement theorem for discrete acts is

given.

In the third chapter the category of acts is defined,

and we establish some of the elementary properties of the

category, such as the coreflectivity of the category of

pairs of sets; from this follows the existence of coproducts.

The most important result in this direction is that any

*Bracketed numbers are references to the bibliography.
1

compact totally disconnected act is the inverse limit of

finite discrete acts. This extends-the-principal result

of [5] from the subcategory of acts with a given input

semigroup to the whole category of acts. Since the inverse

limit of a system of sets is a subset of the cartesian

product of the sets, this result may be considered as an

embedding theorem; hence one may look for conditions under

which each of the finite "approximants" the term is due

to Bednarek and Wallace [5] is of some given cardinality.

The embedding theorem gives conditions which are sufficient

(and for the most part necessary) that a compact totally

disconnected act be embedded in the product of two-point

Boolean lattices acting on themselves via infimum multi-

plication.

The author would like to acknowledge the aid of his

supervisory committee in the preparation of this dissertation;

in particular he is indebted to Professor A. R. Bednarek,

the committee chairman, for many conversations on matters

mathematical. He is also indebted to Alice Anson for her

impeccable typing. This research was supported by the

department of mathematics of the University of Florida and

by the National Science Foundation (GP6505).

1. PRELIMINARIES

1.1 Definitions

A semigroup is a Hausdorff space S with a continuous

binary associative operation defined on it.

An act is a triple (S, X, m) with S a semigroup, X a

topological space and m: S x X X such a continuous function

that, letting sx = m(s,x), s(tx) = (st)x. (We customarily

denote the semigroup operation as well as the function m

by juxtaposition unless the requirement of clarity dictates

otherwise.) We consider m to be fixed once for all and will

refer to the act (S, X, m) simply as (S, X), supressing

explicit mention of m wherever it is possible to do so

without loss of clarity. In addition m will occasionally

be referred to as the action of S on X; S is called the

input and X the state space of (S, X). In the sequel,

(S, X) will always denote an act if no hypothesis to the

contrary is explicit stated.

Of course what is here called an act could have been

called a left act, allowing room for the definition of

the notion of a right act as a triple (X, S, m) where X

is a Hausdorff space S is a semigroup and m: X x S S

satisfies m(m(x,sl),s2) = m(x,sls2) for all x E X and all

sls2 6 S. There is a rather obvious duality, of course,

in the notions, which we formalize briefly. For any

semigroup (S, -), define the dual semigroup to be (S, O)

where s 0 t = t s. Let S' denote the dual of S, supressing

mention of the operation on S. If now (S, X) is a (left)

act, define its dual to be (S, X)' = (X, S', *) where x s =

sx It follows that (S, X)' is a right act. If we make

a similar definition for the dual of a right act (X, S)

as (X, S)' = (S', X, *) with an obvious definition of *

then it follows that (S, X, *)- = (S, X, *) for any act

(S, X). It can be seen that each theorem about (left) acts

is logically equivalent to a "dual" theorem for right acts.

We will rarely have use for this duality (Section 3.4 is

an exception), so it will not be treated in any further

detail.

By convention all sets are topologized with a Hausdorff

topology. The closure, interior and boundary of a set A

will be denoted by A*, A and F(A) respectively; the empty

set is denoted by 0. For ease of expression we note that

the collection P(S) x P(X) of pairs of subsets of S and of

X is a lattice with (A, B) < (C, D) iff A c C c S and

B c D C X. Hence the lattice operations are coordinatewise

union and intersection for join and meet, respectively.

Unless it is specifically stated otherwise, the symbols V

and A will always mean the lattice operations in P(S) x P(X),

i.e. if (A, B) < (S, X) and (C, D) < (S, X) then

(A, B) V (C, D) = (A U C, B U D)

and

(A, B) A (C, D) = (A n C, B n D).

A pair (A, B) in P(S) x P(X) will be called closed iff A = A*

and B = B*. Similar locutions employing other topological

adjectives, such as compact or open for example, are defined

analogously. If (A, B) < (S, X) then AB = {abl (a,b) E A x B).

It is an immediate consequence of the continuity of the

action of S upon X that A*B* c (AB)* for all (A, B) c (S, X),

and A*B* = (AB)* if the action is also a closed function,

which is the case if both S and X are compact.

If for each 6 in a suitable index set A we have

(A6,B6) < (S, X) then it is clear that ( U Ag) U B) =
-EA 6EA
U A 6B and ( n A,) ( B) c n A B ..
(6,6')E/AxA 6EA 6EA (6',6')EAxA

We list a few examples next. Any semigroup S acts on itself

via its multiplication, and any semigroup S acts on any

space X via the identity sx = x. This action, which is just

the projection of S x X onto X satisfies the identities

s(tx) = t(sx) and s(sx) = sx. If, next, X is a locally

compact space then as is well known, the set XX of all

continuous functions from X into itself is a semigroup in

the compact-open topology which acts on X via evaluation:

m(f,x) = f(x) for each pair (f,x) in XX x X. Every topo-

logical transformation group [12]* is an act, as is every

pair (S, I) where I is a left ideal of a semigroup S. In

particular the action of S upon itself via its multiplication

will be denoted by (S, S). If E is a left congruence on S,

*Bracketed numbers are references to the bibliography.

then S acts on S/E in an obvious way ([20], [21], [22], [3]).

A pair (T, Y) < (S, X) is a subact if T is a subsemi-

group of S and TY c Y. The state space of a subact (T, Y) is

elsewhere called a T-ideal ([9], [8]). In particular, we

consider (0, 0) to be a subact.

The product TT (S X ) of a family {(S X ) a 6 A}
EAA

of acts is again an act with the action defined coordinate-

wise: (sx) = s x a E A. It is elementary that the action

so defined is continuous. We will sometimes write

(S1 x S2 x... Sn, X1 x X2 x...x Xn) to mean

n
TT(S,, Xi) if n = 2 or 3.
i=l
A homomorphism from (S, X, m) to an act (T, Y, m') is

such a pair (f,g) that f: S + T is a semigroup homomorphism

(i.e. f is continuous and is algebraically a homomorphism)

and g: X Y is a continuous function and the diagram below

commutes.
m
S x X

fx g m
m
T x Y Y

(The function f x g: S x X + T x Y is given by f x g(s,x) =

(f(s),g(x)) for each (s,x) E S x X.) If (f,g) is a homomor-

phism from (S, X) to (T, Y) we may write

(f,g): (S, X) (T, Y). We remark that if ia : FIS S
aEA
and p : TTX a X are the canonical projection functions
aEA

and (S X ) is an act for each a E A then each pair

(I Pc) is a homomorphism.

If we identify a function with its graph, the (f,g)

notation for homomorphisms is justified.

Proposition 1.1. A homomorphism (f,g): (S, X)- (T, Y)

is a closed subact of (S x T, X x Y)

Proof. If (!s,f(s)),(x,g(x))) is any element of f x g

then (sx,f(s)g(x)) = (sx,g(sx)) is in the graph of g, i.e.

fg c g. The continuity of f and g and the fact that T and

Y are Hausdorff imply that the graphs of f and g are closed,

as is well-known. It is convenient to recall that a con-

gruence C on a semigroup S is an equivalence relation on S

which is a subsemigroup of S x S. We will call a pair

(F, E) admissible on (S, X) if F is a congruence on S, E

is an equivalence on X and (F, E) is a subact of

(S x S, X x X). Proposition 1.3 below appears in [4], but

is repeated here for the sake of completeness. It is a

corollary to a theorem of Sierpinski, part of which we

state here without proof.

Proposition 1.2. (Sierpinski). If, in the

following diagram of topological spaces and con-

tinuous functions,

X -- Y

// h

the conditions

(i) Z = *(X)

(ii) ((x) = ((y) implies 4(x) = 4(y)

(iii) X compact

are satisfied, then there is a unique continuous

function h making the diagram commute. Further-

more if Y = 4(X) and the converse of (ii) holds

then h is a homeomorphism.

Clearly if t and i are homomorphisms of semigroups then

h is also one such. We shall use 1.2 to prove an analogous

result for acts.

Proposition 1.2'. If, in the following diagram

of acts and homomorphisms,

(f2,g2)
(S, X) 2 2 (T, Y)

(fl'g1) -" (hlh2)
(U, Z)

the conditions

(i) fl(S) = U and gl(X) = Z

(ii)' fl(s) = fl(t) implies f2(s) = f2(t) and

gl(x) = gl(y) implies g2(x) = g2(y)

liii)' S and X are compact

then there is a unique homomorphism (hl,h2) making

the diagram commutative.

Proof. The existence of h and h2 follow from two appli-

cations of 1.2; to see that (hl,h2) is a homomorphism, let

(u,z) C U x Z, (u,z) = (fl(s),gl(x)) say. Then hl(u)h2(z) =

(h1f (s)) (h2g1(x)) = f2(s)g2(x) = g2(sx) = h2g1(sx) =

h2 (f (s)gl(x)) = h2(uz).

Proposition 1.3. If (F, E) is a closed admis-

sible pair on a compact or discrete act (S, X)

then there is a unique action of S/F upon X/E for

which the pair (f,g) of natural surjections of S

upon S/F and X upon X/E is a homomorphism.

Proof. Since F and E are closed then, as is well-known,

the compactness (respectively discreteness) of S and of X

imply that the quotient spaces S/F and X/E are compact and

Hausdorff (respectively discrete). It is easy to see that

in the following diagram (i) and (ii) of 1.2 hold with

0 = f g and = gm. Hence there is a unique continuous

m making the diagram commute.

m
S x X X

\ fxg g
m
S/F x X/E X/E

Writing m'(s,x) = s x we see that since m is an action,

f(s) (f(t) g(x)) = f(s) g(tx) = g(s(tx)) = g((st)x) =

f(st) g(x), i.e. f(s) (f(t) g(x)) = f(st) g(x) (1)

Now in order to conclude that m' is an action, we need to

show that S/F is a semigroup with such an operation a that

f(st) = f(s) o f(t). But surely we may apply our result (1)

in the case X = S and E = F. Hence from Sierpinski's theorem

we obtain a function O such that f(s) o (f(t) o f(x)) =

f(st) f(x) and the diagram below commutes

S x S S

f x f f

S/F x S/F S/F

i.e. f(st) = f(s) o f(t). (2)

Hence S/F is a semigroup and from (1) and (2) we may conclude

f(s) (f(t) g(x)) = (f(s) o f(t)) g(x),

i.e. is an action.

10

The act so defined is called the quotient of (S, X)

modulo the admissible pair (F, E).

1.2 Some Isomorphism Theorems

The next lemma is a special case of a categorical

theorem, the general formulation of which would take us

too far afield at present. Another special case, that of

universal algebra, appears as Lemma 11.3.1 in Cohn's

Universal Algebra [7]. We pause to give some notation.

If A, B and C are sets and R is a relation from A to B,
11
i.e. R c A x B, then R1 = {(y,x) (x,y) E R} c B x A and

if also S c B x C then S o R = {(x,y) IZz (x,z) E R and

(z,y) E S}. For any family of sets {Zili E I} the jth

projection from TTfz onto Z. is denoted by r.. If
iEI 3

A' C A and B' c B then A'R = 2((A' x B) n R) =

{yI| x E A' (x,y) E R}; dually RB' = 1((A x B') n R) =

{x|I y E B' .9. (x,y) E R}, and RIA" = R n (A' x A).

Lemma 1.4. Let (Si,Xi), i E (1, 2, 3} be acts

with (T, Y) a subact of (Si, X ) and (Ui, T.) a

subact of (Si x Si+, Xi x Xi+ ), for i = 1, 2.
1 +1 1 +l
Then (TP1, YT1 ), (-, 1) and (2 o Ol 2 o T

are subacts of (S2, X2), (S+ x Si, X.i+ Xi)

and (SI x S3' X1 x X3), respectively.

Proof. Since T2 is a semigroup homomorphism, then

TDD1 2((T x S2) n 1 ) is a semigroup; now if s E TQ1 and

x E Y 1 then for some s' E T and for some x' E Y, (s',s) E &D

and (x',x) E 1i. From the hypothesis, sx E TY c Y and

(s',s)(x',x) = (s'x',sx) E o 'P C L so that sx E YP1'

proving the first assertion. If (sx,ty) = (s,t)(x,y) E

iI i1 then (t,s).E i: and (y,x) E 1, so that (ty,sx) 6
-1
i1 Yi-5 1 whence it follows that (sx,ty) E 1 proving the

second assertion. If now (sx,ty) E (0, o 02 o 2) then

for some u and for some z (s,u) E D1, (u,t) E D2' (x,z) E Y1

and (z,y) E P2. Hence (sx,uz) E ~1 T 5 1b and similarly

(uz,ty) E T2, so that ((2 o ) 2 o ) c5 T o 1 which

proves the third assertion.

Corollary 1.5. (i) The composition of homomor-

phisms is a homomorphism; (ii) The homomorphic image

of an act is a subact.

Proof. In view of 1.1, in the preceding lemma take

(~i' 1) to be a homomorphism.
If f: X Y is a function, its kernel K(f) =

{(x,y) f(x) = f(y)}. The kernel of any function is an

equivalence relation which, furthermore, is closed in X x X

provided that Y is a Hausdorff space and f is continuous;

K(f) is a congruence if f is a semigroup homomorphism.

Proposition 1.6. If (f,g) is a homomorphism

of (S, X) to an act (T, Y), then (K(f), K(g)) is

Proof. If f(s) = f(s') and g(x) = g(x') then f(sx)

f(s)g(x) = f(s')g(x') = f(s'x'), so that K(f)K(g) c K(g);

in view of the immediately preceding remarks the assertion

then follows. We will sometimes use the slightly simpler

notation K(f,g) to denote (K(f), K(g)). A homomorphism

(f,g) is an isomorphism if f and g are bijections. We may

now formulate analogues of the classical isomorphism theorems.

Proposition 1.7. (First Isomorphism Theorem)

If (f,g): (S, X) (T, Y) is a homomorphism of compact

or discrete acts then (f,g) = (ifp, jgo), where

i: f(S) T and j: g(X) Y are inclusion maps,

(p,o): (S, X) (S/K(f), X/K(g)) is the natural

projection and (F,g): (S/K(f), X/K(g)) (f(S), g(X))

is an isomorphism.

(f,g)
(S, X) > (T, Y)

p,o ) (ij)

(S/K(f), X/K(g)) (f(S), g(X))

Proof. 1.6 implies that (K(f), K(g)) is an admissible

pair and then from 1.3 it follows that (S/K(f), X/K(g)) is

an act. 1.5 (ii) implies that (f(S), g(X)) is an act, in

fact a subact of (T, Y), and surely the pair (i,j) of inclu-

sions is a homomorphism. The existence of (f,g) with the

asserted properties follows from 1.2'.

We remark that the factorization of (f,g) into a compo-

sition of injective, bijective, and surjective homomorphisms

is unique up to isomorphism. The proof of this remark is

straightforward but tedious and is omitted, as is the proof

of 1.8 below, which is an analogue of the classical second

isomorphism theorem.

Proposition 1.8. (Second Isomorphism Theorem)

If (T, Y) is a closed subact of a compact or discrete

act (S, X) and (F, E) is a closed admissible pair on

(S, X), then (in the notation of 1.4) (TF, YE) is a

closed subact of (S, X) and

(TF/FlTF, YE/EIYE) = (T/FIT, Y/EIY).

The third isomorphism theorem is an immediate conse-

quence of 1.2'. Recall that (A, B) < (C, D) iff A c C and

B CD.

Proposition 1.9. (Third Isomorphism Theorem)

If (F., E.), i = 1, 2 are closed admissible pairs

on a compact or discrete act (S, X), and if

(F1, El) < (F2, E2), then there is a unique homomor-

phism (f,g) making a commutative diagram with the

natural projections, as in the following diagram.

(S, X) (S/F2, X/E2)

(S/1 (f, X1)
(S/F1, X/E )

1.3 A Schreier Refinement Theorem

A pair (T, Y) < (S, X) is called a left (respectively

right) ideal of (S, X) if T is a left (respectively right)

ideal of S and SY c Y (respectively TX C Y). We say that (T, Y)

is an ideal of (S, X) if it is both a left ideal and a right

ideal. We notice that any (left or right) ideal is a subact.

Associated with an ideal (T, Y) of (S, X) is an admis-

sible pair (F, E) called the Rees pair for (T, Y) given by

F = T x T U AS and E = Y x Y U AX (for any set A, A =

{(x,x) x E Al). Verification that (F, E) is admissible is

routine. If now (S, X) is compact or discrete and (T, Y) is

a closed ideal then the Rees pair for (T, Y) is closed. The

quotient of (S, X) by this admissible pair is called the Rees

quotient and is usually written (S/T, X/Y) rather than

(S/F, X/E). A series from an act (S, X) to a subact (T, Y)

is such a sequence (S., X.), 0 < i < n of subacts of (S, X)

that (So, Xo) = (S, X),(Sn, Xn) = (T, Y) and (Si, Xi) is an

ideal of (Si-1, X i1) for each i = 1, 2,...,n. The Rees

quotients (S i_/Si, Xi_ /Xi) are called the factors of the

series. A refinement of a series (Si, Xi)n= from (S, X)
i=O from

to (T, Y) is such a series (T., Y)j= whose terms include

all the terms of (Si, X.) =. Two series are called iso-

morphic if there is a 1-1 correspondence between them with

paired factors isomorphic as acts.

Theorem 1.10. Any two series from (S, X), a dis-

crete act, to a subact (T, Y) have isomorphic refinements.

Proof. If {(Si, Xi) O < i < n} and {(T., Y.) O < j < p}
1 -
are two series from (S, X) to (T, Y), we define (Sij, Xij) =

(S U (S.i n T.), Xi U (Xi_1 n Y.)) for all (i,j) such that
1 < i < n and O < j < p; dually we define (T j, Yji) =

(T. U (Tj_ n Si), Y. U (Y_ n X)) for all (j,i) such that

1 < j < p and 1 < i < n. Then (S, X) = (S10 X10) (S11, X)11

.- (p, Xlpp = (Sl' X1) (S20' ( 20) ,* Snp' Xnp
(T, Y) and (S, X) = (T0, Y10) (T, Y11) D. . (Tln Y) =
T (TIT Y YT' 20) 2z (T21' Y21) -3 (Tpn Y) = (T, Y)
It is well known [6], [18] that (Si,j) is an ideal of S. ij

for each i,j for each i,j such that 1 < j < p with an obvious

dual remark applying to each T. .. To see that (Si, Xij.)

is an ideal of (Sij-l' Xij-1) it is enough to show

S. Xi c X.. and Sij Xij. c X... The verifications
1,J3-1 ij 1-] 1j Ij-l 1-_
are routine. Since (Si, X.) is an ideal of (Si X. i) for

each i and (Tj, Y.) is an ideal of (TjI, Yj._) we have
S Xi U Si X X. and Tj_ Yj U Tj Y_ C Yj. Then S
1-1 1-1 X a 3-1 3 3-1 ] ij-1
X.. = (Si U (Si-1 n T ))(Xi U (Xi Yj)) = S. X. U

Si(X n Yj) U (SiS1 T j-) Xi U (Si-l n Tj) (Xi-1 n Y.)
Xi U (S.il X T_1 Yj) c Xi U (Xil Yj) = Xj, so

(Sij, X. ) is a left ideal of (S ,j- X. ). That it is
ij ij i,Xj- Tt-i
also a right ideal, and hence an ideal, is verified in a

similar fashion. Now the quotient semigroups S. /Si
1ij-1 ij
and Tji-l/Tj, as is well-known (see [6 ], e.g.), are not

only isomorphic but identical if their respective zero

elements be identified, so that an explicit semigroup iso-

morphism f.. is at hand. Indeed, if O (respectively O')

is the zero of S. /S.. (respectively T. ,/T..) then
nj-l ij 3j,i-l ji

17

Si,j-/Sij = U Si,j-l\ij and Tji-./Tji = O' U T ji_\Tji

But Si,j \Sij = Si \Si n Tj_I\S. n Si,\Tj n Tj_\Tj =

T. ji\Tj. Since this observation is set-theoretic rather
3,i-1 jJ
than algebraic, it follows mutatis mutandis that there is

similarly an explicit one-to-one correspondence gij between

X. /X.i and Yj /Y... It is routine and elementary that
1,j-l 13 ii-1 j1
(f ij,gi ) is an isomorphism.

It is not known if 1.10 holds with "discrete" replaced

by "compact" and "subact" replaced by "closed subact."

1.4 Some Lattices Associated with an Act

Let I(S, X) denote the collection of all nonvoid ideals

of (S, X). Then I(S, X) is itself nonvoid since it contains

(S, X). Now if (S., X.), i = 1, 2, are ideals we have

0 SIS2 S1 n S2l and 0 7 S2X1 SX1 S2X Xl n X2;
furthermore, S(X1 n X2) U (S1 n S2) X c X1 n X2, hence

(S1 0 S2' X1 A X2) is an ideal of (S, X), and is the greatest
lower bound in P(S) x P(X) of (Sl X ) and (S2, X2).

Similarly we see that (S1 U S2, X1 U X2) is an ideal of (S, X)

and is the least upper bound in P(S) x P(X) of (S, X1) and

(S2, X2). It follows that I(S, X) is a distributive sub-
lattice of P(S) x P(X).

Next let A(S, X) denote the collection of all subacts

of (S, X). As before, (S, X) E A(S, X) so this collection

is nonvoid, and since the intersection of any nonvoid

collection of subacts is a subact (we do not require that

the statespace or input of an act be nonvoid), we have that

A(S, X) is a complete infimum semilattice of P(S) x P(X).

Since (S, X) is the least upper bound of A(S, X) and is a

member of A(S, X), then A(S, X) is a complete lattice. The

supremum operation may be defined as (Sl, X) V (S2' X2) =

A {(T, Y) E A(S, X) (SI U S2, X1 U X2)< (T, Y)}. It is clear

that we may define the subact [(A, B)] generated by any pair

(A, B) < (S, X) as above, [A, B] = A {(T, Y) E A(S, X) (A, B) <

(T, Y)}. On the other hand, for A C S let [A] be the

subsemigroup of S generated by A, and for B C X let [B] =

i{Z c XIB c Z and [A] Z c Z}. Then it is easy to see that

[(A, B)] = ([A], [B]). Since [(S1 U S2, X1 U X2)] =

(S1 U S2, X1 U X2) for all ideals (SI, X1) and (S2, X2) it
is clear that I(S, X) is a sublattice of A(S, X).

If we denote by C(S) the lattice of all closed congru-

ences on a compact or discrete semigroup S and by E(X) the

lattice of all closed equivalence relations on a compact or

discrete space X, then the subset C(S, X) of C(S) x E(X)

consisting of all admissible pairs on (S, X) is of some

interest. To see that C(S, X) is a sublattice of

C(S) x E(X), we need only remark that (AS, AX) is a closed

and hence that C(S, X) is a complete infimum semilattice;

since (S x S, X x X) is a maximal element of C(S, X), it

follows that C(S, X) is a complete lattice. Moreover, in

the discrete case we can describe (F1, E ) V (F2, E2) for

any pair (Fi, E.), i = 1, 2 of elements in C(S, X) with the

aid of the following lemma.

Lemma 1.11. If (S, X) is a discrete act and

(F., Ei) E C(S, X) for i = 1, 2 then A (El V E2)

E1 V E2 and (F1 V F2) AX C E1 V E2'

Proof. It is well-known that (x,y) E El V E2 if and

only if there is a finite sequence x = x0, X,... ,xn = y

such that (x.j_,x.) E E1 U E2 for each j, 1 < j < n.

Now if s E S and (x,y) E E 1V E2 then AS E. c F. E. C E.
2 o 1 - 1 1-
for i = 1, 2, hence for each j, (sx j-, sx ) E E U E2,

i.e. (sx,sy) E E1 v E2' proving the first assertion. The

20

second assertion is proved similarly. Now it is clear that

(F- V F2, E1V E2) = (FI, E ) V (F2, E2) for if (s,t) E

F1- V F2 and (x,y) E E V E2 then (sx,sy) and (sy,ty) are in
E V E2 by 1.11, and since El V E2 is transitive, (sx,ty) E

E1 V E2, so (F1 V F2, E1 V E2) is an admissible pair; the

result now follows.

2. RELATIVE IDEALS IN ACTS

We begin by generalizing the notion of ideal. If (T, Y)

is a subact of (S, X), then a pair (A, B) < (S, X) is called

a left (respectively right) (T, Y)-ideal if the following

conditions hold:

TB c B (respectively AY c B)

and

A is a left (right) T-ideal.

(A is a left (right) T-ideal if TA c A, (AT c A) [21], [22],

[3].) (A, B) is a (T-Y) ideal iff it is both a left and a

right (T, Y) ideal. A (T, Y)-ideal will be referred to

generically as a relative act ideal or just relative ideal

if no confusion with the semigroup analogue is possible. In

this chapter we will state and prove some results concerning

the existence of minimal and maximal proper relative ideals.

We remark that an ideal of (S, X) as defined in section 1.3

is an (S, X)-ideal in the present definition. It is further-

more easy to see that the collection of all nonvoid (T, Y)-

ideals for fixed (T, Y) < (S, X) is a sublattice of

P(S) x P(X). The proof parallels that of the analogous

remark in section 1.4. We will use the following notation

adopted from the theory of semigroups ([21], [22], [3]).

If A c S apd',.T is a subsemigroup of S, let

L(A) = A U TA

22

R(A) = A U AT

J(A) = A U AT U TA U TAT.

Juxtaposition here denotes multiplication in S.

If B C X, define

L(B) = B U TB,

juxtaposition now denoting the action of S upon X. It will

be clear from context in which sense L is being used.

2.1 Minimal Relative Ideals

For any pair (A, B) < (S, X) define the left (T, Y)-

ideal generated by (A, B) to be

L(A, B) = A {(P, Q) (A, B) < (P, Q) and (P, Q) is a left

(T, Y)-ideal}.

The right (T, Y) ideal R(A, B) and the ideal J(A, B)

generated by (A, B) are defined analogously.

A (T, Y)-ideal (respectively left ideal, right ideal)

is called minimal iff it is minimal, with respect to the

partial order on P(S) x P(X), among all (T, Y)-ideals

(respectively all left (T, Y)-ideals, all right (T, Y)-ideals).

For the following lemma let (A, B) < (S, X) and let (T, Y) be

a fixed subact of (S, X).

Lemma 2.1. (i) J(A, B) = (J(A), L(B) UJ(A)Y)

(ii) L(A, B) = (L(A), L(B))

(iii) R(A, B) = (R(A), B U R(A)Y)

Proof. We prove only (i); the remaining assertions are

proved similarly. It is clear that J(A, B) is a (T, Y)-ideal

containing (A, B); if on the other hand (P, Q) is any (T, Y)-

ideal containing (A, B) then J(A) c p since P is an ideal of

S, and furthermore B U TB U J(A)Y c Q U TQ U PY c Q so that

J(A, B) < (P, Q). The assertion (i) is hence proved. The

next result is the act analogue of a basic result in semi-

groups [13].

Theorem 2.2. If (T, Y) is a closed subact of

a compact act (S, X) then there is a unique minimal

(T, Y)-ideal; the state space and input of this minimal

ideal are closed.

Proof. Let (C, <) be the collection of all closed (T, Y)-

ideals (i.e. both the input and the state space are closed

sets), with the partial order inherited from P(S) x P(X).

(T, Y) is such a closed (T, Y)-ideal, hence C is nonvoid.

If C" is any chain in C and if (P, Q) = AC' (recall that

infimum in P(S) x P(X) is coordinatewise set intersection),

then P and Q are each the intersection of a chain of closed

sets, so by the compactness of S and of X, P* = P \$ 0 i

Q = Q*. If now (F, G.) 6 C" then (P, Q) < (F, G), hence

TQ U py c TG U FY c G; since (F, G) is arbitrary, it follows

that TQ U py c Q, i.e. (P, Q) is a closed (T, Y)-ideal, and

is the lower bound of the chain C'. Since we have proved

that every nonvoid chain in C has a lower bound, we may con-

clude from Zorn's lemma that C has a minimal element, (J, K)

say. If now (A, B) is any (T, Y)-ideal, not necessarily

closed, and if (A, B) < (J, K) then for any point (s,x) E

A x B we have first that J(s) = {s} U s T U Ts U Ts T C A

since A is a T-ideal and next that L(x) U J(s) Y = x U Tx U

J(s) Y C B U TB U AY C B; but (J(s), L(x) U J(s) Y) is the

ideal generated by (s,x), and is closed; hence (J, K) <

(J(s), L(x) U J(s) Y) < (A, B) < (J, K). It follows that

(J, K) = (A, B), i.e. (J, K) is minimal among all (T, Y)-

ideals. We observe next that (J, K) < (T, Y) since evi-

dently (T I J,. Y n K) is a (T, Y)-ideal and therefore

contains (J, K). In particular, J is a semigroup for

J2 c TJ c J. To see that (J, K) is unique as asserted let

(J', K') be any minimal (T, Y)-ideal; then 0 # JJ' c TJ' n

JT c j' n J, so J 0n J is a T-ideal. Furthermore 0 #

(J n J')Y c JY n J'Y C K n K" and hence T(K n K') U

(J n J')Y c (TK n TK') U K n K' c K n K' so (J n J', K n K')

is a (T, Y)-ideal and is contained in both (J, K) and (J', K')

The minimality of the latter then imply J = J N J' = J' and

K = K n K' = K' so (J', K') = (J, K), proving uniqueness.

We remark that the uniqueness does not depend on any topo-

logical hypothesis on (S, X). It is easy to see that in

any act (S, X) if (P, Q) is the minimal (T, Y)-ideal for any

(T, Y), then P is necessarily the minimal T-ideal ideal of

(S, X). Indeed if P' is a T-ideal contained in P, then

TQ U P'Y c TQ U PY c Q, so that (P', Q) is a (T, Y)-ideal,

and hence P = P'. Conversely if S has a minimal T-ideal J

then (J, JY) is surely a (T, Y)-ideal, for T(JY) U JY c

(TJ)Y U Jy c JY. If now (P, Q) is a (T, Y)-ideal and

(P, Q) < (J, JY) then P = J by minimality and Q c JY. On

the other hand QD TQ U PY so it follows Q = JY, proving

that (J, JY) is a minimal (T, Y)-ideal. In summary, we have

proved the following proposition.

Proposition 2.3. An act (S, X) has a minimal

(T, Y)-ideal iff S has a minimal T-ideal; if J is the

minimal T-ideal, then (J, JY) is the minimal (T, Y)-

ideal.

We remark that if S is a semigroup with a subsemigroup

T and a minimal T-ideal J then the minimal (T, T)-ideal of

(S, S) is (J, J). For by hypothesis JT c J, and from 2.3

26

it follows that (J, JT) is the minimal (T, T)-ideal, so we

need only show J c JT. But this is trivial since JT is a

T-ideal and hence contains the minimal T-ideal J.

If now (S, X) is any act and (S, X) E S x X, define

Sl(s) = {sn n > 1)* and 2(s,x) = (x} U {snxln > 1)*. From

the continuity of the action it follows that rF(s)F2(s,x) C

r2(s,x) with equality holding if the action of S upon X is

a closed function, e.g. (S, X) is a compact act. Hence

(Fl(s), F2(s,x)) is a subact of (S, X). From theorem 2.2,

if (Fl(s), T2(s,x)) is compact, it has a minimal ideal (K, J)

and since Fl(s) is an abelian semigroup K must be a group;

(K, J) is hence a compact group acting on a compact space.

2.2 Maximal Proper Relative Ideals

Maximal proper ideals in semigroups were studied by

Koch and Wallace [14].

If (A, B) < (S, X) and C c X we define

A[-]B = {x E X Ax C B}

A(-B = {x E X Ax n B 3 0}

BC[-11 = {s sjsC c B}

BC(-1) = {s E S sC n B # 0)

The notation will frequently occur in the sequel. The

following facts are well-known and proofs are available in

many places, e.g. [2], [5].

Lemma 2.4. Let (A, B) < (S, X)

(i) If A is compact and B is closed then A(-)B

is closed

(ii) If A is compact and B is open then A[ B

is open

(iii) If B is open then A (-B is open

(iv) If B is closed then A[-]B is closed.

Similar assertions of course hold for BC[1 and

BC(-) for B, C X..

Lemma 2.5. If (T, Y) is a subact of (S, X) and

(T, J) is a subact of (S, X) then JY1[-l is a T-ideal.

Proof. If s E J[-1] then sY J. If also t E T then

(ts)Y = t(sY) tJ c TJ c J so ts E JY[11. Also (st)Y =

s(tY) c s(TY) c sY c J so st E JY[-1], proving the assertion.

For (A, B) < (S, X) the largest (T, Y)-ideal contained

in (A, B) is (M(A), M(A, B)) = V {(P, Q) (P, Q) is a (T, Y)-

ideal and (P, Q) < (A, B)}. Since M(A) in particular is a

T-ideal it is contained in M(A), the maximal T-ideal con-

tained in A. We give next a condition for which M(A) =

M(A).

Proposition 2.6. In the above notation, a

necessary and sufficient condition that M(A) = M(A)

is that M(A) C M(A, B)Y

Proof. (->) It is no loss to suppose (A, B) contains

at least one (T, Y)-ideal for if not it follows that M(A) = 0

and the proposition holds. So we suppose M(A) 7 0 7 M(A, B),

and since (M(A), M(A, B)) is a (T, Y)-ideal, then T M(A) U

M(A) Y C M(A, B) so that M(A) c M(A, B)Y[]. Hence the

condition is necessary. To prove sufficiency, we use the

hypothesis to conclude T M(A, B) U M(A)Y c M(A, B) and

hence we deduce that (M(A), M(A, B)) is a (T, Y)-ideal,

which implies (M, A), M(A, B) < (M(A), M(A, B)). In partic-

ular M(A) c M(A) and we are finished.

Lemma 2.7. (i) If (A, B) = (A*, B*) then

M(A) and M(B) are closed.

(ii) If (S, X) is compact, (T, Y) is a closed

subact of (S, X) and (A, B) = (AO, B) then M(A)

and M(A, B) are open sets.

Proof. (M(A)*, M(A, B)*) < (A*, B*) = (A, B) implies

M(A)* c M(A) and M(A, B)* c M(A, B), proving the first

assertion. To prove the second assertion, suppose (t,x) E

M(A) x M(A,B). We may assume without loss of generality

that there is a (T, Y)-ideal (P, Q) such that (t,x) E P x Q.

Indeed from the definition of (M(A), M(A, B) there are (T, Y)-

ideals (P, Ql) and (P21 Q2) contained in (A, B) such that

t t P1 and x E Q2. But (P1 U P2' Q1 U Q2) is also a (T, Y)-

ideal contained in (A, B), and (t,x) E (P1 U P2) x (Q1 U Q2).

Since (P, Q) is a (T, Y)-ideal then J(t,x) < (P, Q), i.e.

J (t) = t U Tt U tT U TtT c_ P A = AO and J2 (y) = y U Ty U

J (t) Y c Q C B = B. By repeated application of Wallace's

theorem [13] to the various compact sets whose union is

Jl(t) and J2(y) it follows that there are open sets U about
t and V about x such that J(U, V) < (A, B) so that J(U, V) <

(M(A), M(A, B)). Since (U, V) < J(U, V) it follows that

t E U c M(A) and x E V c M(A, B); since t and x were chosen

arbitrarily, we have shown that M(A) and M(A, B) are open.

We say that a pair (A, B) is properly contained in (S, X),

(A, B) < (S, X) if S A x X B # 0.

Theorem 2.8. If (T, Y) is a closed subact

of a compact act (S, X) then each proper (T, Y)-

ideal is contained in a maximal suchand each maxi-

mal proper (T, Y)-ideal is open.

Proof. If (P, Q) is a proper (T, Y)-ideal and

(s,x) E S\P x X\Q, then (P, Q) < (M(S\s), M(S\s, X\x)), an

open proper (T, Y)-ideal which proves that the partially

ordered collection (B, <) of open proper (T, Y)-ideals con-

taining (P, Q) is nonvoid. Let (K, J) be the supremum of a

maximal chain C = {(C D) IX E A} in B, i.e. (K, J) =

V (CX, D ); (K, J) is seen to be an open (T, Y)-ideal.
XCA X

If (K, J) is not proper then either K = S or J = X, which

implies, via the compactness of S and X, that S or X, as the

case may be, is the union of finitely many open proper sets

C1 ,...,Cx (respectively D ,...,D ). But the inputs as
1 n 1 n

well as the state spaces are linearly ordered since C is,

hence the union is just one of the CL (respectively, Dx ),
i i

which is absurd. Hence (K, J) is a proper ideal, and is

clearly a maximal such. That (K, J) is open follows from

lemma 2.7 via the equation (K, J) = (M(S\s), M(S\s, X\x))

which holds for any (s,x) E S\K x X\J.

Proposition 2.3 is a description of the relation between

the minimal ideal of S and the minimal (T, Y)-ideal of (S, X),

viz, the existence of either implied the existence of the

other, and the former is necessarily the input semigroup of

the latter. Now in the case of maximal proper ideals, the

situation is more complicated, in part because the existence

of a maximal proper ideal does not imply its uniqueness, as

is well-known from semigroup theory. Hence in light of 2.3,

(1) If K is a maximal proper T-ideal of S, when does

there exist some J c X such that (K, J) is a

maximal proper (T, Y)-ideal?

(2) Conversely, if (K, J) is a maximal proper (T, Y)-

ideal, when is K a maximal proper T-ideal of S?

is not known but lemma 2.9 sheds some light on the matter.

For convenience we adopt a term used by Paalman-de Miranda [17]

and say that an act (S, X) has the maximal property relative

to a subact (T, Y) if each proper (T, Y)-ideal is contained

in a maximal such.

Proposition 2.9. If (S, X) has the maximal

property relative to a subact (T, Y) and K is a

maximal proper T-ideal of S such that KY X then

there is some J c X such that (K, J) is a maximal

proper (T, Y)-ideal.

Proof. The hypotheses on K together with the trivial

inequality KY U T(KY) C KY imply that (K, KY) is a proper

(T, Y)-ideal and hence, by the maximal property, is contained

in some maximal proper (T, Y)-ideal (P, J). But then K c p

so the maximality if K implies K = P, proving the assertion.

Lemma 2.10. If (K, J) is a maximal proper

(T, Y)-ideal and K' is a maximal proper T-ideal

of S such that K c K" then

(i) K = K' <=> K' c JY[-l] and

(ii) K = JY [1 or JY-11 = S.

Proof. Evidently if (K, J) is a (T, Y)-ideal then K =

K C JY[-. Conversely if K' c JY then (K', J) is a

proper (T, Y)-ideal which contains the maximal (T, Y)-ideal

(K, J). Hence K = K'. For the second assertion we merely

note that since JY[-l] is a T-ideal and (JY[-l])Y U TJ c J,

that (JY1[-l, J) is a (T, Y)-ideal; evidently K c JY-1

so if K JY[-1] the maximality of (K, J) implies JY[-l = S.

Theorem 2.11. Let (T, Y) be a closed subact

of a compact act (S, X) and denote by E the set of

idempotents of T. If (K, J) is a maximal proper

(T, Y) ideal such that E c K, then TY c J.

32

Proof. Suppose to the contrary that for some (t,y) E

T x Y, ty E TY\J. Necessarily y J. It is clear that

(K, J U Ty) is a (T, Y)-ideal and that J is a proper subset

of J U Ty. Hence the maximality of (K, J) implies J U Ty =

X, and we may conclude from this that y E Ty, i.e. y = sy

for some s E T. Since the set yy(-1) = {p E S|y = py} is

closed and contains sn for all n > 1 then it contains the

compact subsemigroup F(s) = {snn > 1}* and hence contains

the idempotent of F(s), i.e. y = ey, where e = e E F(s).

But ey E EY c KY c J by hypothesis so y E J, which is

absurd.

3. CATEGORICALITIES

We wish to prove a representation theorem which generalizes

a result of Bednarek and Wallace [5] on inverse limits of

finite machines. The generalization is most easily seen when

it is couched in the language of categories and it is for

this reason that we digress to include the necessary concepts

from the theory of categories. A systematic exposition of

this theory may be found in Mitchell [15] and a more compact

exposition may be found in Freyd [11]. The reader who is

familiar with the definition of a category may omit section

3.1.

3.1 Categories

A category G is an entity consisting of two classes 8

and 7, not necessarily sets, with ? a disjoint union of the

form U HomG[A, B] satisfying the following axioms.
(A,B) MxM

(1) Hornm[A, B] is a set for each pair (A, B) E6 x .

(2) For each triple (A, B, C) E 0 x O x 0 there is a

function called composition from HomG[B, C] x

Hom (A, B) into Hom [A, C] whose value at each point

(a,B) is denoted by Ba, and which satisfies

(i) (yB)a = y(Ba) whenever both sides of this

equation are defined;

(ii) For each A E 0 there is an element 1A E

Hom (A, A) satisfying 1 A = a and 1A =

whenever the composition of (1A, a) or (8, 1B)

is defined.

The members of 0 are called objects of G and the members

of M are called morphisms of G. When there is no ambiguity

we write Hom [A, B] instead of Hom [A, B]. If a E Hom [A, B]

we call A the domain and B the codomain or range object of

a, we write A = dom a and B = codom a. The fact that

a E Hom [A, B] is represented schematically by A -> B or

a: A --> B.

It is easy to see that the morphism 1A is unique for

each A and hence we may unambiguously call it the identity

morphism of A.

Certain types of morphisms are distinguished in a

category. We define some of these next. A morphism a

of a category G is called a monomorphism of G (or is said

to be monic) if whenever ax = ay then necessarily x = y; it

is an epimorphism of G (or is epic) of xa = ye implies x = y;

it is an isomorphism of G if there is a morphism B of the

category such that B: X A = dom a, aS = 1x and Ba = 1A.

We remark that an isomorphism is always both monic and epic

but the converse need not hold.

A category G is a subcategory of a category C if each

object of G is an object of C and if Hom (A, B) is a subset

of Hom (A, B) for each pair (A, B) of objects of G. One

must notice that the property of being epic or monic depends

on the category being considered. It can be shown, e.g.

that epimorphisms in the category of topological spaces

(See example (ii) below) are the continuous surjections

while in the subcategory of Hausdorff spaces the epimorphisms

are the dense maps (a map f: X Y is dense if f(X) is a

dense subset of Y), while in the sub-subcategory of compact

Hausdorff spaces the epimorphisms are again the continuous

surjections.

Examples of the categories abound:

(i) The category 8 of sets in which the objects are

sets and a is in Homg[A, B] if and only if a is a function

from A into B. We must, in order to verify that 8 is a

category, distinguish between a: X Y and a: X Z in the

case Y c Z and a(x) = a(x) for each x E X. For otherwise

the class of morphisms is not a disjoint union of sets of

morphisms. Such distinction will be assumed and rarely -

if ever again made explicit. Note that a: X Y is not

necessarily onto, of course.

(ii) The category G of topological spaces has the class

of all topological spaces for objects and all continuous

functions for morphisms. (The composition of continuous

functions is again continuous, so the verification of the

axioms 1 and 2 is routine.)

(iii) The category of discrete semigroups has for objects

all discrete semigroups (i.e. semigroups S endowed with the

discrete topology) and all homomorphisms for morphisms. The

composition of homomorphisms is a homomorphism.

(iv) The category of topological semigroups or more

simply of semigroups has all semigroups for objects (recall

that the continuity of the semigroup operation is included

in the definition of semigroup; see page 3 ) and all contin-

uous semigroup homomorphisms for morphisms. Again continuity

is part of the definition of semigroup homomorphism, so this

category will be referred to simply as the category of semi-

groups.

Notice that the category of discrete semigroups is a

subcategory of the category of semigroups, and that the class

of compact Hausdorff spaces determines a subcateogry of the

category of topological spaces; in this subcategory the

monomorphisms are just the continuous injections, the epi-

morphisms are the continuous surjections and the isomorphisms

are the bicontinuous bijections, i.e. homeomorphisms. It

is also the case that the class of compact semigroups determine

a subcategory of the category of semigroups. Here the mono-

morphisms, epimorphisms and isomorphisms are, respectively,

the one-to-one homomorphisms, surjective homomorphisms and

the iseomorphisms (an iseomorphism is both a homeomorphism

and an isomorphism). We remark that the category of discrete

semigroups has an epimorphism which is not onto, namely the

inclusion map of the natural numbers (zero included) into

the integers, with addition the semigroup operation in both

cases. The result follows from the fact that any homomor-

phism of the integers is determined by its restriction to

the nonnegative integers. A characterization of the epimor-

phisms in the category of semigroups is not known to this

writer. A topological example of this phenomenon is

furnished by the inclusion map of the rationals into the

reals (usual topologies) which is an epimorphism in the

category of Hausdorff spaces and continuous functions.

An exposition of category theory may be found in the

books by Mitchell [15] and Freyd [11] to which the reader

is directed for a further knowledge of the theory.

Proposition 3.1. The class of all acts and

the class of all homomorphisms between acts are

respectively the object class and morphism class

of a category.

Proof. We remark first that an act is not completely

specified until the sets S and X, together with a semigroup

structure on S and an action of S on X, are given. So a

homomorphism (f,g): (S, X) (T, Y) is not completely

specified until the semigroup structures on S and T as well

as the two actions are given. It is now seen to be the case

that the class of homomorphisms is a disjoint union as

required by the definition of category. It is clear that

the collection of homomorphisms between any two acts is a

setso axiom 1 is satisfied. From 1.5 (i) we see that part

(i) of axiom 2 is satisfied if we define the composition

(f,g) o (T,g) of two homomorphisms to be (ff,gg) whenever

this makes sense. We may take the pair (ls, 1X) of identity

functions to satisfy part (ii) of axiom 2.

In the sequel this category will be called C for the

sake of brevity. It will be convenient also to distinguish

the subcategory Co whose objects are compact acts and whose

morphisms are those of C, i.e. (f,g) is to be a morphism of

Co iff (f,g): (S, X) (T, Y) and (S, X) and (T, Y) are both

compact acts.

Proposition 3.2. Suppose (f,g): (S, X) (T, Y)

is a homomorphism. Then (f,g) is an isomorphism in C

iff f is an iseomorphism and g is a homeomorphism.

Proof. If (f,g) is an isomorphism then for some (f,g)

we have (ls, lX) = (f,g) o (f,g) = (ff,gg) (1)

and (1T, ly) = (f,g) o (C,g) = (ff,gg) (2)

so that f and g are invertible functions, from which fact

the necessity follows. If, conversely, (f,g) is a homomor-

phism and f is an iseomorphism and g is a homeomorphism
-1 -1
then the pair (f ,g ) is a morphism of the category and

satisfies equations (1) and (2) which proves that (f,g) is

an isomorphism of C.

No such characterization of the monomorphisms or the

epimorphisms of C is known.

39

We proceed next to investigate the existence in C of

certain categorical constructs as prolegomena to a study of

compact totally disconnected acts.

3.2 Products of Acts

Let G be a category and {A : A E A} a set of objects of

G. The product of the family is such a pair (A, {I7: A E A})

that A is an object of G, T : A AX for each A E A and such

that if B is any object in a and fx: B AA is given for each

SE6 A then there exists a unique f: B A, called the product

morphism, such that fX = 7 f for each A 6 A, i.e. the diagram

below commutes for each A E A:

f
B------*> A

AA

Now if (A, {rX: X E A}),(A', (PX: A E A}) are products of

the family {A : X E A} then two applications of the definition

yield unique morphisms fl: A A' and f2: A' A such that

IT = P fl and p = T f2 for each A, so that I = r f2fl'

Now fi = lA,' and the uniqueness of the product morphism

allows us to conclude f2fl = 1A. Similarly one shows that

flf2 = 1A, which proves that A and A' are isomorphic. These

results are well-known and it is for this reason that they

are not stated as propositions here. We do, however, have

the following result.

Proposition 3.3. The direct product of acts

defined in section 1.1 is the product in C.

Proof. Let {(SX, X )| E A} be any collection of acts.

The direct product, as defined in section 1.1, of this

family is the act ( TTS, TXTX) where the action is coordin-
XEA XA

atewise, i.e. (sx)X = sAx for each s E S = TSx each
AEA

x C X = 17Xi and each A E A. Now suppose (T, Y) is an act
XEA A

and {(f ,g) : (T, Y) (S X) I1 E A} is a family of homomor-

phisms. Define f: T S and g: Y X by (f(t)) = fx(t)

and (g(y))x = g (y) for each X E A, i.e. p f = f and

oGg = g, for each X E A where p, and a, are the canonical

projections of S and X onto S and TA respectively. Now

for each E6 A, (p 0X) is a homomorphism from the way the

action of S on X is defined, namely o (sx) = p (s)o (x) for

all s E S, all x E X and all X C A. Then it follows that

(f,g) is a homomorphism, for if t, s E S and x E X then

p f(ts) = f (ts) = f (t)f (s) = p (f(t))p (f(s)) = px(f(t)f(s)),

i.e. f(ts) = f(t)f(s) and

oag(ts) = gx(ts) = f (t)g (s) = p (f(t))oC(g(s)) = off(t)g(s),

i.e. f(ts) = f(t)g(s).

It is elementary that f and g so defined are continuous and

are unique with respect to making the diagrams below commute

for each X E A.

f g
T ----- S Y X

Sx XX

42

Hence (f,g) is the unique homomorphism making the diagram

below commute.

(f,g)
(T, Y) (S, X)

(f('gX (pa)
(SV, X )

3.3 Coproducts and Free Acts

The definition of coproduct or direct sum is dual to

that of product in the sense that one is obtainable from

the other by "reversing all the arrows" in either definition.

We will defer exposing the definition however until we have

discussed free acts [7], for the existence of the latter is

needed to prove the existence of the former.

In the category of discrete semigroups, a free semigroup

on a set A is a pair (w, F(A)) with w such a function and

F(A) such a semigroup that, if S is any semigroup then any

function f: A S extends uniquely to a homomorphism

F: F(A) S so that the diagram below commutes.

A F (A)
7--
f/

S

It is well-known that any two free semigroups on A are iso-

morphic and that a free semigroup exists on every set:

let F(A) = U An with the operation given by
n=l

(al,... ap) (bl,...bq ) = (al,... ap,bl ....bq)

for all al, ...a bl,... b E A all positive integers p and

q; w:A F(A) is w(a) = (a). This particular representation

F(A) is called the canonical free semigroup on A, and leads

us to a representation of the notion of free act.

Let now (A, B) be any ordered pair of sets. The

free act on (A, B) is an act (F(A), F(A, B)) together with

a pair of functions ml: A F(A) and w2: B F(A, B) such

that if (S, X) is any act and (f,g): (A, B) + (S, X) is

any pair of functions, then there is a unique homomorphism

(f,g) making the diagram below commutative.
(Wll 2)
(A, B) 2 > (F(A), F(A, B))

(fg) (f,g)

(S, X)

Proposition 3.4. In the category of discrete

acts, a free act exists on every pair of sets.

Proof. Let F(A, B) = (F(A) x B) U B where F(A) is the

canonical free semigroup on A. We define a function

F(A) x F(A, B) + F(A, B) as follows: If s E F(A), s =

(al,....a ) say and b E B then s b = ((al,...a ),b).

If s is as above and x = ((xl,...x ),b) E Aq x B then

s x = ((al,...a ,x,...x ) ,b). It is easy to see that

* is an action. Take (i,j): (A, B) + (F(A), F(A, B)) to

be i(a) = a and j(b) = b for each a E A and each b E B.

Suppose now that (f,g): (A, B) (S, X) is any pair

of functions from (A, B) into (S, X). Since F(A) is the

free semigroup on A, there is exactly one homomorphism

f: F(A) S such that fi = f. Define g: F(A, B) + X by

g(b) = g(b) for b E B and g((x1,...x ),b) = T(xl,...x q)g(b)

for all ((X1 ...x ),b) E F(A) x B. It can be seen that

(f,g) is a homomorphism. In fact, if (al,...a ) E F(A)

and ((bl,...b q),b) E F(A) x B then f(al ... a )g((b ... bq) ,b) =

f(al... ap) (f(b ,...b q)g(b)) = f(al,... a ,bl ....b )g(b) =

( (a .. .ap,b ...b ) ,b) = g((al, . a )*((bl ... bq) ,b));

also if b E B and (al,...a ) E F(A) then f(al,...a )g(b) =

f(al,...a )g(b) = g((al,...a ),b) = g(al,...a) b). Hence

(f,g) is a homomorphism and (f,g) o (i,j) = (f,g), proving

the proposition.

Evidently if A = B then F(A, B) = F(A), and the free

act on (A, A) is the free semigroup F(A) acting on itself.

One can prove, just as for semigroups, that every act (S, X)

is isomorphic to a quotient of a free act; the free act in

question is just (F(S), F(S, X)) and it follows that (S, X)

is the image of this free act and hence (S, X) is isomorphic

to a quotient of (F(S), F(S, X)). We can now discuss the

notion of coproduct in the category of discrete acts.

If {AxIA E A} is any family of objects in a category C,

the coproduct of the family is any object A together with a

family of morphisms {i : Ax + AIX E A} having the property

that if X is any object in G and fx: Ax X is a morphism

for each x E A, then there is exactly one morphism f making

the diagram below commute for each x E A:

f
X A-------A

A X
A

The uniqueness of-f implies that any two coproducts of

{AXIX E A} are isomorphic. Proof of this assertion is

similar to that of the similar assertion for products and

is omitted. We use IIAI to denote any coproduct of

{AXI E A}.

Proposition 3.5. Coproducts exist in the

category of discrete acts.

Proof. Let {(S., X ) A E A} be any family of discrete

acts; it is no loss of generality to suppose that

S, S = 0 = X n X whenever A P y, so we do make this

supposition. We also define S = U S, X = U XA, and
AEA AEA

define pX: S, S by p,(s) = s for all s E SX,

qX: X +- X by q,(x) = x for all x E X If now (T, Y) is
a discrete act and (f ,g ): (SA XA) (T, Y) is a homomor-

phism for each A E A, then define f: S T and g: X Y by

f(s) = f (s) if s E SX and

g(x) = g (x) if x E XA.

These functions are well-defined since the S.'s and the X 's

are both pairwise disjoint families. Letting (i,j) be the

natural injection of (S, X) into (F(S), F(S, X)) we have

that (f,g) extends to a unique homomorphism

(f,g): (F(S), F(S, X))- (T, Y) by proposition 3.4.
Now define the sets Fo and Eo as follows:

Fo = {(z,w) E F(S) x F(S) 1X A (x,y) E S x SX pIp(xy) =

z and ip (x)iTp(y) = w}. Eo = {(a,b) E F(S, X) x F(S, X)

2 A E A (s,x) E S x XX * jq (sx) = a and

ip,(s) Jq,(x) = b}. Since Fo C F(S) x F(S) and
Eo C F(S, X) x F(S, X), the collection of all admissible

pairs (P, Q) > (Fo, Eo) is not empty; let (F, E) denote

the infimum of all such admissible pairs and let (p,q) be

the canonical homomorphism from (F(S), F(S, X))to

(F(S)/F, F(S, X)/E). Referring to the commutative diagram

below, we verify next that E c Ker g.

(1,j)
(S, X) >-(F(S), F(S, X))
(pX ,q) /

(fl'g) / (f,g)

(T, Y) (F(S)/F, F(S, X)/E

Indeed if (a,b) E E then for some X C A there is some

(s,x) 6 S xX such that

a = jq,(sx) and b = ipT(s) jq,(x).
Hence g(a) = g(jq (sx)) = (gj)q (sx) = gq (sx) = g,(sx) =

f (s) gC(x) = fpx(s) gq,(x) = (fT)p,(s) (gj)q (x) =

f(Tp (s)) g(jq (x)) = g(Tp (s) jq (x)) = g(b),
i.e. E c Ker g.

It follows mutatis mutandis that F c Ker f, so that

(F, E) < K(f,g); since p and q are surjections, the hypo-
thesis of 1.2' is satisfied allowing one to conclude the
existence of a unique homomorphism (f,g): (F(S)/F, F(S, X)/E) -

(T, Y) satisfying

(fp,gq) = (f,g).
In summary we have shown, letting (i,j) = (pipX,qjqi)
that there is a unique (f,g) making the diagram below

analytic for each A E A, proving that (F(S)/F, F(S, X)/E) =

A (Sx' X )-.

(i,j)
(S X ) (F(S)/F, F(S, X)/E)

(fXg) (T (f,g)
(T, Y)

3.4 Inverse Limits of Compact, Totally Disconnected Acts

We wish to formulate-a-sharper-version of a result of

Bednarek and Wallace (Theorem 1, [5]) concerning the state

space of a compact totally disconnected act. It is necessary

first to define the notion of inverse limit in a category.

We recall that a quasi-ordered set (A,<) is a set A

with a relation < on A which is reflexive and transitive.

If A < p it is sometimes convenient to write p > A. A

quasi-ordered set (A,<) is-called directed if for each

(A,U) E A x A there is some v E A -9- A < v and p < v.

An inverse system (A,,p ,A,<) in a category G is a

family of objects {AIA E A} indexed by a directed set

(A,<) together with a family {p IX > y} of morphisms,
A A
p A A such that p = 1 and if X > 1 > v, then the
p AX A _

diagram below commutes:

A X--p A
p

A V

A

The inverse limit in G of an inverse system (AX,p ,A,i)

is an object L of G together with a collection

{ irF: L AilA E A} of morphisms of G such that if X is any

object of G and {f : X A IA E Al is any family of morphisms

of G such that p f = f whenever A > y, then there is

exactly one morphism f making each diagram with A > p

commute:

f
X ---- L

f \
A

The uniqueness of f implies that the inverse limit of any

inverse system is unique up to isomorphism. The proof of

this assertion is quite similar to the proof of uniqueness

of the direct product and is omitted. It is occasionally

convenient to write lim AX for the inverse limit of a system
A<--
(A,,p ,A,<). If (A,,p ,A,<) is an inverse system in the

category of compact Hausdorff spaces then lim A =
<--A

{x 6 TA I A > ===> p n(x) = T (x)} and is a closed,
XAEA i

non-empty subset of TTAA. These assertions are proved in
ACA

Eilenberg and Steenrod, Chapter VIII [10].

If S is a compact semigroup then the class of all

compact acts (S, X) are the objects of a category G(S), the

morphisms of which are homomorphisms of the form (lsg).

The result of Bednarek-Wallace alluded to above can be

stated now as follows.

Theorem 3.6. (Bednarek-Wallace). A compact

act (S, X) with X totally disconnected is the inverse

limit in G(S) of acts (S, X ) having finite, discrete

state spaces.

The techniques used to prove this theorem may be applied

to prove the following result.

Theorem 3.7. If (S, X) is a compact, totally

disconnected act (i.e. both S and X are compact and

totally disconnected) then (S, X) is the inverse

limit in the category of compact acts of finite

discrete acts.

The theorem will be proved from the following sequence

of lemmas. The references below are to Numakura [16] and

Bednarek and Wallace [5]. We recall that for any set Y

that the diagonal of Y is Ay = {(y,y) y E Y}; A is closed

in Y x Y if and only if Y is a Hausdorff space.

Lemma 3.8. (Numakura). If Y is a compact

Hausdorff space and E is an open equivalence relation

on Y then Y/E is finite and discrete.

Lemma 3.9. (Numakura). If Y is a compact,

totally disconnected space and U is an open subset

of Y containing A then there is an open equiva-

lence relation E on Y contained in U. Moreover

if Y is additionally a semigroup then there is an

open congruence E c U.

It is well-known (See, e.g. [5]) that an open equiva-

lence relation on a Hausdorff space is necessarily closed,

i.e. clopen.

Lemma 3.10. (Bednarek-Wallace). If (S, X)

is a compact act, X is totally disconnected and

V is an open subset of X x X containing the diagonal

of X, then there is such a clopen equivalence E c V

that (As, E) is an admissible pair.

Lemma 3.11. If (S, X) is a compact totally

disconnected act and U and V are such open subsets

of S and of X respectively that A U and A c V,
s x-
then there is a-clopen admissible pair (F, E) <

(U, V).

Proof. We may deduce from an application of 3.10 that

there is a clopen equivalence relation E c V such that

A E C E. Since E is in particular open then the continuity

of the action implies via Wallace's theorem that there is

an open set Uo containing the compact set As which has the

property that UoE C E. An application of 3.9 to the semi-

group S implies the existence of a clopen congruence F C

Uo n U. Now FE C UoE C E, proving that (F, E) is an admis-

sible pair. Since (F, E) < (U, V) the lemma is proved.

We return now to the main result of this section.

Proof of 3.7. If (S, X) is a compact, totally discon-

nected act then 3.11 implies that the family 7 of all clopen

admissible pairs (F, E) < (S x S, X x X) is a nonvoid

family; clearly is a filter base relative to the partial

order < on P(S) x P(X) (i.e. if (F E l)and (F2, E2) are in

9 then there is some (F3, E ) < (Fi, E1) (F2, E2); we

may take F = F 0 F2 and E = E1 N E2). Furthermore,

AJ = (As, A ). Indeed, if (s,t) E S x S\As and (x,y)

X x X\AX, then the regularity of X x X implies the existence

of an open set V c X x X such that AX C V and (x,y) { V.

Similarly there is an open set U such that A c U and

(s,t) ( U. Then apply 3.11 to obtain (F, E) E 3 with

(s,t) ( F and (x,y) E.

Now we index 3 by a suitable set A and, observing that 3 is

quasi-ordered by <, define a relation < on A by declaring

A < p if and only if (F E ) < (F E ). Since 3 is a

filter base it follows that (A,<) is a directed set. If

we denote the canonical pair of quotient surjections of

(S, X) upon (S/F X/Ex) by (p,,q) then wheneverA > p

it follows from Sierpinski's theorem (1.2') that there is
A X
a homomorphism (f ,g ) making the diagram below commute.

(S, X)

(PXqX) (Ppq)
} ((fA gA
(S/F, X/E)- ---> (S/F, X/E )

Evidently f= f g = g if A > p > v, and
A X
(fg ) = (I/F, 1/E for all A E A. Now let Sm

(S E6 S/FJA > W --> fX 7T(s) = T (s)}, (i.e. S is the

canonical representation of lim S/FA) and define Xm similarly.

It is easy to check that (S5, X_), considered as a subact

of ]7(S/FA, X/EQ), is the inverse limit in the category of
AEA

compact acts of the inverse system (S/FA, X/E ) of acts.

Indeed if (f ,g ): (T, Y) (S/Fx, X/Ex) is given for any
act (T, Y) and all A A and satisfies (f gA )(fg)

(f ,g,) whenever A > p, then there- is a semigroup homomor-

phism f: T S. and map g: Y Xm satisfying 7Xf = f and

P =g = g9 for all A, where mA and pA are the projection
functions from S/FA onto S/FA and from TT X/E, onto
AXA AZA

X/Ex respectively. We show first that (S X_) is a subact

of the product. If A > p and (s,x) t S x X then gp (sx) =

g A(T (s)p (x)) = (f (s)) (g (x)) = 7 (s)P (x) = P (sx),
since (f ,g ) and (r,,p,) are all homomorphisms. Hence

SmXm X To see that (f,g) is a homomorphism, let t E T
and y E Y. Then p g(ty) = gA(ty) = f (t)gx(y) =

iTf(t)p g(y) = pC(f(t)g(y)) for each A E A, i.e. g(ty) =

f(t)g(y). Hence we have proved the assertion that (Sm, X) =

lim (S/F,, X/Fx). Now in particular there is a homomorphism

(f,g): (S, X) (S., X.) satisfying iAf = p and pg =

q. for each A E A. Since A7 = (AS, AX), then f and g

separate points of S and X respectively. It follows now

that f and g are homeomorphisms, so that we may conclude

from 3.2 that (f,g) is an isomorphism; this observation

concludes the proof of 3.7.

3.5 An Embedding Theorem

If (S, X) is such an act that s(tx) = t(sx) for all

s,t E S and all x E X, we say that S acts commutatively on

X; if s(sx) = sx for each s E S and each x E X, we say S

acts idempotently on X; if S acts both commutatively and

idempotently on X we call the action trellis-like, and say

S acts trellisly on X [5]. The action of S on X is called

effective if, for each pair of distinct elements s,t in S

there is some x E X such that sx 3 tx. Notice that a

semigroup may act on itself either commutatively or idem-

potently without being a commutative semigroup or a semi-

group each of whose elements is idempotent. For example,

if S is any Hausdorff space define st= t for all (s,t) E

S x S. Then S is a semigroup and s(tx) = tx = x = sx =

t(sx) but S is not commutative if it contains at least two

elements; if next to E S is fixed and we define st = to for

all (s,t) t S x S then any t t to is not idempotent, but

for all s,t,x E S we have t(tx) = to = tx. Of course, if

S is commutative (respectively, idempotent) then S acts

commutatively (respectively, idempotently) on any space on

which it acts. However, if S acts effectively on X then S

acts trellisly iff S is a semilattice. The proof is an

immediate consequence of the fact that there is a canonical

embedding b of S into XX, the semigroup of all functions

from x to itself, given by [s(s)](x) = sx for each s C S

and each x t X; 0 is evidently a one-to-one homomorphism.

Hence if S acts commutatively, '(S) is a commutative sub-

semigroup of XX, i.e. S is commutative; a similar remark

holds if commutativee" is everywhere replaced by "idempotent."

We recall that a semilattice is a commutative, idem-

potent semigroup (in the sense in which commutativee" and

"idempotent" are usually used in algebra). By the pre-

ceding remark any action by a semilattice is necessarily

trellis-like.

A prime right (left) ideal of an act is such a right

(left) ideal (A, B) that if tx e B then either t ( A or

x E B. A prime ideal is both a prime left and prime right

ideal. The definition can be made for prime relative ideals,

of course, but we will have need only of the notion in the

"absolute" case. For any act (S, X), the index of an

admissible pair (F, E) is an ordered pair (p,q) of cardinal

numbers such that p is the cardinal number of S/F and q is

the cardinal number of X/E.

In the paper "Finite Approximants of Compact Totally

Disconnected Machines" [5], the following theorem was proved.

Theorem 3.12. (Bednarek-Wallace) If (S, X) is

a compact trellis-like act with X = Sz for some

z E X, and X is totally disconnected, then there

is an isomorphism (1 ,@) of (S, X) into an act

(S, B) where B is a compact boolean lattice.

(Theorem 3.12 is only a part of theorem 2 of [5];

according to a theorem of Aczel and Wallace,[1], [23],

such a state space X must support the structure of a semi-

group and then the remainder of theorem 2 of [5] asserts

that 0 is a semigroup homomorphism into the A-semilattice

of the boolean lattice B.) We state next the principal

theorem of this paper, and then a sequence of lemmas needed

for its proof.

Theorem 3.13. If (S, X) is a compact, totally

disconnected, effective trellis-like act such that

neither card S nor card X is 1 and x E Sx for each

x E X then (S, X) can be embedded in a direct product

of acts (S., X ) where, for each X, (S,, XX) is one

of the following two acts:

(i) (Sx, XX) is the semilattice of order 2 acting

on itself via its multiplication

(ii) SX is trivial and the action satisfies the

identity tx = x for each x E X .

The proof of 3.13 is deferred until the following lemmas

have been proved; the first two parts of 3.14 are from [5],

but the remaining results seem to be new.

Lemma 3.14. If S acts trellisly on X and x E Sx

for each x e X then

(i) the relation P = f(x,y) x t Sy} is a partial

order on X, called the natural order on X

(ii) AT E C E if E = (X\S(-l)a x X\s(-l)a) U

(S(-l)a x S(-l)a) for any element a E X

(iii) letting V: X X/E be the canonical sur-

jection, then P = [T x ] (P) is a partial

order on X/E.

We remark that in the proof of (ii) one uses the fact that

(S, X S(-1)a)is a left ideal of (S, X). The proof of (iii) is

immediate since P is'the natural order on X/E, and the quo-

tient act (S, X/E) itself satisfies the hypothesis of lemma 1.

Lemma 3.15. Let (S, X) be an act with such an

element x in X that S\xX(-1) is not empty. Then

S\xX(-) is (i) a right ideal of S which is prime

if S acts idempotently; (ii) an ideal if S acts

commutatively

Proof (i). Suppose to the contrary that t E S\xX(-)

s E S and ts E xX(- Then for some x, E X we have

x = (ts)xo = tsxo, from which it is concluded that

t E xX(-1), contrary to the supposition t f xX(-1)

Hence S\xX ) is a right ideal of S. If now p and q

are in xX(-1) then x E pX N qX, so that x = pyl = qy2 for

some yl, y2 E X; since the action is idempotent, x =

p(pyl) = p(qy2) = (Pq)Y2' which implies that pq E xx(-l)
so that xX(-l) is a semigroup. But this means that

S\xX(- is prime.

(ii) If now S acts commutatively and S\xX(-1) is not

a left ideal then there is some s E S and some t E S\xX(-1

such that st E xx(-l) which is to say x = stxo for some

xo E X. But stxo = tsxo, so that x E tX; this implies

t E xX(-1), which is absurd.

Lemma 3.16. If I is a prime ideal of a semi-

group S or I = 0 then (I x I) U (S\I x S\I) is a

congruence on S.

The proof is omitted. The following special case wil

be needed in the sequel.

Corollary 3.17. If (S, X) is a trellis-like

act and x t X then F = (S\xX(-1) x X\x(-1) U

(x ) x(-) is a congruence on S.
(xX xX ) is a congruence on S.

1

)

We remark that S\xX(-l) may be empty so that F = S x S.

For example if X = S is a semilattice acting on itself via

its multiplication and having a zero element x then xX(-)

S.

Lemma 3.18. Let (S, X) be a trellis-like act

and x E X. Then

(i) (S\xX(-1) (S(-l)x) c X\S(-l)x

(ii) S(X\S(-l)x) c X\s(-1)x

(iii) (xX(-I) S (-l)x) S(-1)x

Proof. (i) If s E S\xX(-1) then x E X\sX; if also

y E S(-)x and sy E S(-)x, then x E S(sy) = s(Sy) c sX,

(ii) If y E X\S(-l)x and if for some s E S, sy E s(-l)x

then x E Ssy c Sy, i.e. y E S(-)x, a contradiction.

(iii) If s E xX(-) and y E S(-)x then for some xl 6 X

and some sl E S we have x = sx = s y. Then x = sx1 =

s(sxl) = sx = s(sly) = sl(sy) E Ssy, which is to say sy E

S -)x, proving (iii).

Corollary 3.19. Let (S, X) be a trellis-like

act and x E X; take F as in 3.17 and E as in 3.14 (iii)

with a = x; then (F, E) is an admissible pair.

Proof. The only thing not already verified is that

FE C E, but this follows from the definitions of F and E

and an application of 3.18.

In the sequel let 2 = {O,1} be the semilattice with

O < 1 and let 1 = {1} be a subsemilattice of 2. We define

(2,2) to be the action of 2 on 2 via semilattice multipli-

cation:

0 1

0 00

1 01

and (1,2) is understood as a subact of (2,2). Its multipli-

cation table is

0 1

1 0 1

Lemma 3.20. Let (S, X) be an effective trellis-like

discrete act such that a E S(-l)a for each a E X, and

suppose (s,x) and (t,y) are distinct points of S x X.

Then there is an admissible pair (F, E) on (S, X) of

index (1,2) or (2,2) such that (<(s), 4(x)) j

(4(t), f(y)) where ( S,) : (S, X) (S/F, X/E) is the

canonical morphism. Furthermore, (i) if (F, E) has

index (1,2) then (S/F, X/E) is isomorphic to (1,2);

(ii) if the index of (F, E) is (2,2) then (S/F, X/E)

is isomorphic to (2,2).

Proof of (i). If (s,x) (t,y) then either x / y, or

x = y but s 3 t. We consider first the case x Z y and we

may suppose without loss of generality that y S(-l)x.

(Indeed if y E S(-1)x and x E S(-l)y then x E Sy and

y E Sx, which implies via 3.14 (i) that x = y.) Apply

3.18 to the point x to obtain the admissible pair (F, E).

Since (S, X) is discrete, (S/F, X/E) is an act and since

y E x S(-1)x and x E S(-1)x by hypothesis, then *(x) \$ 4(y).

In this case the hypothesis of effectivity is not used, a

fact of particular importance in the sequel. In case x = y

and s I t, then the effectivity hypothesis implies that

sxo A txo for some Xo E X. As before, it is no loss of

generality to suppose that txo S(-1)sxo, and with this

assumption we claim that t f (sxo)X(-). For otherwise

sxo E tX, and hence there is some x, t X such that

sxo = tx1, implying sxo = txl = t(txl) = t(sxo) = s(txo) E

S(txo), or equivalently that txo S(-l) (sxo), contrary to

assumption. Evidently s t (sxo)X(-) so the admissible pair

(F, E), obtained by an application of 3.18 to the point sxo,

is of index (2,2). Since s t (sxo)X(-I) and t t S\(sxo)X(-1)

we have 4(s) A f(t), so the first part of the lemma follows.

The isomorphism assertions are immediate in view of 3.18.

Proof of 3.13. We suppose now that (S, X) is a compact,

totally disconnected, effective, trellis-like act such that

a E Sa for each a E X, and that (s,x) and (t,y) are distinct

points of S x X. We wish to prove that there is a homomor-

phism (4,f) such that (f(s), p(x)) \$ (f(t), f(y)). There

are two cases to consider: either x A y, or x = y and

s A t.

(I) x A y. We have A X X x X\{(x,y)} = V a proper

open subset of X x X and if we choose any proper open subset

U of S x S, lemma 3.11 applies to produce a clopen admis-

sible pair (F,, Eo) < (U, V), so that (S/Fo, X/Eo) =

(So, Xo) is a finite discrete and trellis-like act such

that a E Soa for each a E Xo, and furthermore x A y where

in general t is the equivalence class in Xo containing the

element t of X. (The same notation will be used for the

equivalence classes in So.) Now since x y y, as noticed

earlier the hypothesis of effectivity is not needed to deduce

from 3.20 that there is an admissible pair (F, E) of index

(1,2) or (2,2) on (So, Xo) such that the equivalence classes

in Xo containing x and y are distinct. Now if (t,P) is the

completion of the following diagram, we have W(x) \$ f(y),

so case I follows.

canonical canonical
(S, X) > (So, Xo) Z-(So/F, Xo/E)

( b) isomorphism

S(2,2)

(Of course if the index of (F, E) is (1,2) the isomorphism

is to (1,2) which is a subact of (2,2); this inclusion is not

shown on the diagram.)

(II) x = y and s \$ t. Since the action of S on X is

effective, there is some w E X such that sw i tw. We assume

without loss of generality that tw S(-1)(sw) and hence

conclude as in the proof of 3.20 that t \$ (sw)X(-). Now

(A s, A) < (S x S\(s,t), X x X\(sw,tw)) and since (S, X)
is a compact and totally disconnected act we apply 3.11 to

deduce the existence of a clopen admissible pair (Fo, Eo) <

(S x S\(s,t), X x X (sw, tw)) making (So, Xo) = (S/Fo, X/Eo)

a finite discrete act, which is trellis-like. Since tw 4

S(-) (sw) and sw 6 S(-1)sw, we have tw X sw, and in this

case the effectivity hypothesis in 3.20 is not needed to

conclude that there is a clopen pair (F, E) on (So, Xo) of

index (1,2) or (2,2) such that (sw,tw) \$ E. We claim that

in addition (s,t) t F. For F induces the decomposition

{So\(sw)Xo(1), (sw)Xo '} of So, and since we have already

observed that t E So\(sw)Xo we need only see that

s swXo(-); but this is equivalent to saying sw E sXo which

is certainly the case. Hence F has index 2 and, letting

(t,W) be the completion of the following diagram, we have

proved #(s) # W(t); hence case II follows.

canonical canonical
(S, X) (So, Xo) ------ (So/F, Xo/E)

( isomorphism

~ (2,2)

To conclude the proof let (P, Q) = (2,2). Since
(s,x) (t,y)

for each copy of (2,2) there is a homomorphism (C,t):

(S, X) (2,2) by the above constructions, then if we apply

3.3 we will obtain a homomorphism (f, ) : (S, X) (P, Q).

Now if x and y are distinct points of X it follows from (I)

that 9(x) f ((y) so that 7 is 1-1; similarly T is 1-1, so

that (I, ') is an isomorphism into (P, Q) and 3.13 is proved.

There is an easy corollary to 3.13 which is almost

immediate.

Lemma 3.21. If (S, X) is such an idempotent act

that X = Sz for some z E X, then x E Sx for each x E X.

Proof. For if x E X then for some s E S, x = sz = s(sz)

sx E Sx.

Lemma 3.22. If (S, X) is trellis-like and satisfies

X = Sz for some z E X and (F, E) is an admissible pair

of index (1,2) or (2,2) then necessarily the index of

(F, E) is (2,2).

Proof. If to the contrary F = S x S then suppose (x,y) E

X x X. By hypothesis there is some (tl,t2) E S x S such that

(x,y) = (tlz,t2z) = (tlt2) (z,z) E FE E E,, implying that the

index of E is 1, which is absurd.

Corollary 3.23. If (S, X) is a compact, totally

disconnected, effective, trellis-like act such that

X = Sz and neither S nor X has cardinal number 1,

then (S, X) may be isomorphically embedded in a

product of acts (2,2) such that each factor is the

homomorphic image of (S, X).

Proof. In view of 3.21, the hypotheses of 3.13 are all

satisfied; if we now note that each admissible pair (F, E)

constructed in the proof of 3.13 must have index (2,2) by

3.22, and that the separating homomorphism (4,p): (S, X) -

(2,2) is surjective on both coordinates in each case, the

corollary follows.

APPENDIX

Actions of Order (2,2)

The order of an act (S, X) is a pair (p,q) of cardinal

numbers such that p = card S and q = card X.

Let S = {0,1} and X = {a,b} be;considered just assets.

It is well-known [19] that there are only four non-isomorphic

semigroups on the set S; we denote these as Sl, S2, S3, S4

and give their Cayley tables below.

0 1 0 1 0 1 0 1

0 0 1 0 00 0 01 O 00

1 0 1 1 00 1 1 0 1 0 1

S1 S2 S3 S4

In this appendix we summarize the computation of all actions

of Si upon X for each i, i = 1, 2, 3, 4. The actions are

conveniently given by their Cayley tables where tx appears

as the entry in the t-row and x-column of the table. The

sixteen functions S x X > X,l < j < 16 are given below.

ab

O aa aa aa ab

1 aa a b bb ab

(1) (2) (3) (4)

b a a a b a ab

a b ba b a b a

(5) (6) (7) (8)

ab bbb a a b

aa aa aa bb

(9) (10) (11) (12)

ba bb bb bb

bl b a ab bb

(13) (14) (15) (16)

In si.eral cases, there is an obvious one-to-one corres-

pondence between some S. and X which makes X into a semi-

group isomorphic to S in such a way that the action of S.

upon X is just semigroup multiplication; we will refer to

this phenomenon by saying the action copies the semigroup's

multiplication. For convenience we say Si acts constantly

on X if tx = x0 for all t,x and some fixed xo E Xi.. S. acts

protectively if tx = x for all t,x. A trivial action is

either constant or projective.

In the following summary, (Si,.j) denotes the act

(Si, X, X, ij.

(I) Trivial actions

(Si, p.) for all i and j = 1, 4, 16.

(S p4), (S2' 1i), (S2' K16) copy multiplication.

(II) Non-trivial actions

(S3, u8) copies multiplication

(S4', 2) and (S4, c15) copy multiplication and are
isomorphic.

For each remaining j, we exhibit an s,t 6 S. and an x E X
I

such that (st)x f s(tx).

In S,
In SI,

In S,

S3, 1 (1
In S,
1(1

In S1,

In S, S
In '

10:
Ull:

U12:
V13:

14:

15:

We write s x

S3, O(1-b) r (Ol)b; In S2,

S2, S4, 1(0-a) I (l-O)a; In

2', S3' S4, O(O-a) \$ (O-O)a

4, l(1la) \$ (1-1)a; In S2, 2
*b) X (11l)b

S2, S3, S41 O(O-a) ) (O-O)a

14, l(l-a) # (ll1)a; In S2'

2,' S4, 1(Ob) ? (l-O)b; In

S2' S4, l(O-a) f (l-O)a; In

S2' S3, S4, O(O-a) / (O-O)a

S2, S4, 1(O-a) f (l-O)a; In

S2, S3, S4, O(O-a) ? (O-O)a
S4, fl-lb) 3 (11)b; In S2,

O(l-a) 7 (O-1)a; In S2, S3,

= j (s,x).

L(1-b) X (1-1)b

3', l(l-a) f (1-l)a

L(Oa) \$ (l-O)a; In

L(O-a) 7 (1-O)a

S3, O(lb) ( (O-l)b

S3 0(1la) (O-l)a

S3, l(l-a) 7 (1-1)a

S3, 1(1-a) # (1-1)a

1(l-a) # (1-1)a.

BIBLIOGRAPHY

[1] Aczel, J. and Wallace, A. D. A Note on Generalizations
of Transitive Systems of Transformations, Colloq.
Math (XVII) Fasc I 1967 pp. 29-34.

[2] Bednarek, A. R. and Norris, E. M. Congruences and Ideals
in Machines (Submitted).

[3] Bednarek, A. R. and Wallace, A. D. Relative Ideals and
Their Complements, Rev. Rom. de Math. Pures et App.
(XI) 1966 pp. 13-22.

[4] __ ,quivalences on Machine State Spaces,
Mathematicky Casopis (17) 1967 pp. 3-9.

[5] Finite Approximants of Compact Totally
Disconnected Machines, Math. Syst. J. (T) 1967
pp. 209-216.

[6] Clifford, A. H. and Preston, G. B. Algebraic Theory of
Semigroups, Mathematical Surveys Number 7, American
Math. Soc. 1961.

[7] Cohn, P. M. Universal Algebra, Harper and Row, New York
1965.

[8] Day, J. M. and Wallace, A. D. Semigroups Acting on
Continue, J. Aust. Math Soc. (VII) 1967 pp.327-340.

[9] Multiplication Induced in the State
Space of an Act, Math. Syst. J. (I) 1967 pp. 305-314.

[10] Eilenberg, S. and Steenrod, N. Foundations of Algebraic
Topology, Princeton University Press, Princeton,
N. J. 1952.

[11] Freyd, P. Abelian Categories, Harper and Row, New York
1964.

[12] Gottschalk, W. H. and Hedlund, G. A. Topological
Dynamics, AMS Coll. Pub., Vol. 36, Providence, 1955.

[13] Hoffman, K. H. and Mostert, P. S. Elements of Compact
Semigroups, Charles E. Merrill Books, Columbus,
Ohio 1966.

[14] Koch, R. J. and Wallace, A. D. Maximal Ideals in
Compact Semigroups, Duke Math. J. (21) 1954
pp. 681-685.

[15] Mitchell, B. Theory of Categories, Academic Press, New
York 1965.

[16] Numakura, K. Theorems on Comoact Totally Disconnected
Semigroups and Lattices, PAMS (8) 1957 pp. 623-626.

[17] Paalman-de Miranda, A. B. Topological Semigroups,
Mathematisch Centrum, Amsterdam 1964.

[18] Rees, D. On Semigroups, Proc. Camb. Phil. Soc. (36)
1940 pp. 387-400.

[19] Tetsuya, H., et. al. All Semigroups of Order at Most 5,
J. of Gakugei Tokushimu University (VI), December
1955 pp. 19-39 plus an unpaginated errata sheet.

[20] Wallace, A. D. Retractions in Semigroups, Pac. J.
Math. (7) 1957 pp.1413-1417.

[21] Relative Ideals in Semigroups I,
Colloq. Math. (9) 1962 pp.55-61.

[22] Relative Ideals in Semigroups II,
Acta. Math. Acad. Sci. Hung. (XIV) 1963 pp. 137-148.

[23] Recursions with Semigroup State Spaces,
Rev. Rom. de Math. Pures et App. (XII) 1967
pp. 1411-1415.

BIOGRAPHICAL SKETCH

Eugene Michael Norris was born July 4, 1938, in New

York City. In June, 1956, he was graduated from Hillsborough

High School in Tampa, Florida. A one-semester residence at

the University of Tampa followed, to be succeeded by a two-

year stint as an electronics technician on the Atlantic

Missile Range. During 1959 he attended the University of

Florida and then returned to his position as electronics

technician on Ascension Island and Grand Bahama Island.

From March 1961 until September 1962 he was employed by the

National Aeronautics and Space Administration at Goddard

Space Flight Center. He entered the University of South

Florida in September 1962 and received the degree of Bachelor

of Arts in mathematics in August 1964. Since that time he

has been in the department of mathematics at the University

year 1965-1966, as well as the present year, as an interim

instructor.

Eugene Michael Norris is married to the former

Lois McGibney. He is a member of the Mathematical Association

of America and of the American Mathematical Society.

This dissertation was prepared under the direction of
the chairman of the candidate's supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Arts and Sciences
and to the Graduate Council, and was approved as partial
fulfillment of the requirements for the degree of Doctor
of Philosophy.

June, 1969

Dean, College/ f Artf and Sciences

Supervisory Committee:

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