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
TABLE OF CONTENTS
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 twosided 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 acttheoretic 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 extendstheprincipal 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 twopoint
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 compactopen 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 Tideal ([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 wellknown. 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 wellknown,
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) =
{xI 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 Yi5 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
an admissible pair.
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 (Si1, 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 11 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 (Sijl' Xij1) it is enough to show
S. Xi c X.. and Sij Xij. c X... The verifications
1,J31 ij 1] 1j Ijl 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
11 11 X a 31 3 31 ] ij1
X.. = (Si U (Si1 n T ))(Xi U (Xi Yj)) = S. X. U
Si(X n Yj) U (SiS1 T j) Xi U (Sil n Tj) (Xi1 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 Tti
also a right ideal, and hence an ideal, is verified in a
similar fashion. Now the quotient semigroups S. /Si
1ij1 ij
and Tjil/Tj, as is wellknown (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
njl ij 3j,il ji
17
Si,j/Sij = U Si,jl\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 settheoretic rather
3,i1 jJ
than algebraic, it follows mutatis mutandis that there is
similarly an explicit onetoone correspondence gij between
X. /X.i and Yj /Y... It is routine and elementary that
1,jl 13 ii1 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
admissible pair which is contained in every admissible pair
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 wellknown 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) Tideal.
(A is a left (right) Tideal if TA c A, (AT c A) [21], [22],
[3].) (A, B) is a (TY) 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 Tideal 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 Tideal. 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 Tideal ideal of
(S, X). Indeed if P' is a Tideal 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 Tideal 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 Tideal; if J is the
minimal Tideal, then (J, JY) is the minimal (T, Y)
ideal.
We remark that if S is a semigroup with a subsemigroup
T and a minimal Tideal 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
Tideal and hence contains the minimal Tideal 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 wellknown 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 Tideal.
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
Tideal it is contained in M(A), the maximal Tideal 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 wellknown from semigroup theory. Hence in light of 2.3,
we may ask two questions:
(1) If K is a maximal proper Tideal 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 Tideal of S?
Proposition 2.9 answers question (1); an answer to (2)
is not known but lemma 2.9 sheds some light on the matter.
For convenience we adopt a term used by Paalmande 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 Tideal 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 Tideal
of S such that K c K" then
(i) K = K' <=> K' c JY[l] and
(ii) K = JY [1 or JY11 = 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 Tideal and (JY[l])Y U TJ c J,
that (JY1[l, J) is a (T, Y)ideal; evidently K c JY1
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 Sy = 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 subsubcategory 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 onetoone 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 wellknown 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 wellknown 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 AA
A X
A
The uniqueness off 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 welldefined 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 formulateasharperversion 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 quasiordered 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
quasiordered set (A,<) iscalled 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 Xp 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
nonempty 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 BednarekWallace alluded to above can be
stated now as follows.
Theorem 3.6. (BednarekWallace). 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 wellknown (See, e.g. [5]) that an open equiva
lence relation on a Hausdorff space is necessarily closed,
i.e. clopen.
Lemma 3.10. (BednarekWallace). 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 aclopen 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
quasiordered 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 trellislike, 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 onetoone 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
trellislike.
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. (BednarekWallace) If (S, X) is
a compact trellislike 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 Asemilattice
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 trellislike 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 trellislike
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 trellislike 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,
a contradiction.
(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 trellislike
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 trellislike
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, trellislike 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 trellislike 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 trellislike. 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 11; similarly T is 11, 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 trellislike 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, trellislike 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 wellknown [19] that there are only four nonisomorphic
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 trow and xcolumn 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 onetoone 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) Nontrivial 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(1b) r (Ol)b; In S2,
S2, S4, 1(0a) I (lO)a; In
2', S3' S4, O(Oa) $ (OO)a
4, l(1la) $ (11)a; In S2, 2
*b) X (11l)b
S2, S3, S41 O(Oa) ) (OO)a
14, l(la) # (ll1)a; In S2'
2,' S4, 1(Ob) ? (lO)b; In
S2' S4, l(Oa) f (lO)a; In
S2' S3, S4, O(Oa) / (OO)a
S2, S4, 1(Oa) f (lO)a; In
S2, S3, S4, O(Oa) ? (OO)a
S4, fllb) 3 (11)b; In S2,
O(la) 7 (O1)a; In S2, S3,
= j (s,x).
L(1b) X (11)b
3', l(la) f (1l)a
L(Oa) $ (lO)a; In
L(Oa) 7 (1O)a
S3, O(lb) ( (Ol)b
S3 0(1la) (Ol)a
S3, l(la) 7 (11)a
S3, 1(1a) # (11)a
1(la) # (11)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. 2934.
[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. 1322.
[4] __ ,quivalences on Machine State Spaces,
Mathematicky Casopis (17) 1967 pp. 39.
[5] Finite Approximants of Compact Totally
Disconnected Machines, Math. Syst. J. (T) 1967
pp. 209216.
[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.327340.
[9] Multiplication Induced in the State
Space of an Act, Math. Syst. J. (I) 1967 pp. 305314.
[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. 681685.
[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. 623626.
[17] Paalmande Miranda, A. B. Topological Semigroups,
Mathematisch Centrum, Amsterdam 1964.
[18] Rees, D. On Semigroups, Proc. Camb. Phil. Soc. (36)
1940 pp. 387400.
[19] Tetsuya, H., et. al. All Semigroups of Order at Most 5,
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[20] Wallace, A. D. Retractions in Semigroups, Pac. J.
Math. (7) 1957 pp.14131417.
[21] Relative Ideals in Semigroups I,
Colloq. Math. (9) 1962 pp.5561.
[22] Relative Ideals in Semigroups II,
Acta. Math. Acad. Sci. Hung. (XIV) 1963 pp. 137148.
[23] Recursions with Semigroup State Spaces,
Rev. Rom. de Math. Pures et App. (XII) 1967
pp. 14111415.
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 onesemester 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
of Florida as a graduate student, having spent the academic
year 19651966, 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
Dean, Graduate School
Supervisory Committee:
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