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On metric lattices

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On metric lattices
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Lehman, Alfred Baker, 1931-
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1 v. : ; 28 cm.

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Algebra ( jstor )
Conceptual lattices ( jstor )
Copyrights ( jstor )
Distance functions ( jstor )
Integers ( jstor )
Mathematical transitivity ( jstor )
Mathematics ( jstor )
Partially ordered sets ( jstor )
Semigroups ( jstor )
Sufficient conditions ( jstor )
Dissertations, Academic -- Mathematics -- UF ( lcsh )
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Mathematics thesis Ph. D ( lcsh )
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Thesis - University of Florida.
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Bibliography: leaves 69-71.
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Manuscript copy.
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Vita.

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This item is presumed in the public domain according to the terms of the Retrospective Dissertation Scanning (RDS) policy, which may be viewed at http://ufdc.ufl.edu/AA00007596/00001. The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator(ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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ON METRIC LATTICES By ALFRED BAKER LEHMAN A Dissertation Presented to the Graduate Council of The University op Florida In Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA JANUARY, 1954

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r ACKN01-7LEDGEMEIITS The writer wishes to acknowledge the assistance of his supervisory conimittee and in particular Professors David Ellis, Gaines Lajig, and Robert Blake. « 11

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TABLE OF COITTEOTS Page ACiaTOT'JLEDGMI^irrS ii IHTRODUCTION 1 CHAPTER I. PRELIMINARY CONCEPTS 3 II. PSEUDO-LINEAR QUADRUPLES 19 III, TIIE TRAITSPORI'iATION u(x) = |x| + r(a,x) 35 IV. THE DETERiaHANT j j (a ,a ) | |^ ^3 V, ADDITIONAL PROBLEMS 59 TABLE OP a^; 1 < 1 < 2^ 67 GLOSSARY OP SPECIAL SYMBOLS 6g BIBLIOORAPHY 69 BIOGRAPHICAL SiCETCH 72 ill

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1 INTRODUCTION In modem mathematics, the term "algebra" pertains to a set under finite operations and relations, and "topology" under infinite ones. A lattice (1, 13, 19), a type of partially ordei^ed set, is usually classified as an algebra, althougli it may be associated vxith many topologies [22),^ A metric space (3, 17), based on the notion of distance, defines a highly usefia topology. A metric or normed lattice is a structure combining • the properties of a lattice with those of a metric space, first studied by Gllvenlco (g, 9, 10) and von Neumann (1J5), Although metric lattices may be developed by adding lattice structure to certain metric spaces, it is more natural to impose metric structure on suitable lattices; the associated lattices being more familiar tlian the associated spaces. On the other hand a characterization of metric spaces associated vrlth metric lattices Is known (7, 12), while a characterization for associated lattices is not (see Chap. V), Metric lattices have applications: in the distributive case to the theory of measure (20), integration (27), and probability (1); in the complemented case to projective geometries (1, IS); and in the chain condition case, to the n ^^^^f^"^-^! Pfiren theses refer to the bibliography concluding the dissertation. -1-

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-2theory of groups (11). In these applications, however, the lattice structure is primary; then the norm or dimension function; and least, if at all, the norm-derived metric structure. This v7ork is concerned chiefly x^ith some algebraic and geometrical properties of metric lattices. Although some of the results have possible extensions to sjialysis, the importfjit phase of topological properties has been omitted. The preliminary concepts are somewhat self-contained, assuming only a laiowledge of elementary algebra and an acquaintance with set theory. With space limitations sjid specific applications to metric lattices in mind, some of the usual concepts have been modified or eliminated. As the reader progresses a symbolism is developed and proofs become less detailed, Hrjiy of the results were originally obtained through a variety of methods. Haase diagrams (1, 13) and free lattices (30, 31) were frequently used but not included in this work. Inasmuch as no effective decision procedure is knovm for free modular lattices (2, 23, 2g, 31) , one finds it necessary to employ geometric, arithmetic, and algebraic methods.

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CHAPTER I PRELIMINAHY CONCEPTS Convention 1.1. The dual symbols }, ][, g will be used to denote two expressions; one using the lovxer symbols pjid the other using the upper S'lnbols at eaoh occurence of a dual symbol. The symbol <> used for J will denote the conJunction of the upper and lox^r parts of the equation Involved. Convention 1.2. For each symbol A( ) defined the symbol A[ ] is defined to mean A( ) with the additional condition that formally distinct symbols inside the brackets represent imequal elements. Convention I.J.. a, b, c, d, e, f, h are set elements; 1, j, k, X, m, n are non-negative integers; q, r, s, t are real numbers; u, v, w are real valued functions on a lattice; ajid x, y, z are set elements quantified over the structure concerned. Convention lA, Unless other^Tlse specified all free subscripts run from 1 to n, ajid superscripts from 1 to 2^-1, Convention 1.^. iJhen used in connection vrith real numbers, | | denotes absolute value and | | || denotes a determinant. { } denotes a set. Convention 1.6. iThen not explicitly i::iven, the structures Involved may be inferred from context and the symbols present. -3-

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Deflnltlon 1,1, a quasi-ordered set is a set tocether with a binary relation, <, satisfying: 3C < X (reflexivity) (2) X < y ajid y < z imply x < z (transitivity) Convention 1,2. a < b if and only if b > a. Definition 1.2. A binary relation defined on the pairs of a set is an equivalence relation if (reflexivity) (2) x^y implies y-^x (symmetry) (3) x-^^y ajid y-A^z implies x z (transitivity) Corollary 1.1. In a quasi-ordered set the binary relation defined by a ^ b if and only if a < b sjid b < a is an equivalence relation. Definition 1.^, A partially ordered set (poset) is a quasl-oiviered set under < satisfying: (3) X < y 8jid y < X imply x = y Convention l.g. a $ b if and only if a | b and a h. Definition 1.4, A simply ordered set is a poset satisfying: (4) X < y or y < x (simple order) Convention l.£. a * b if neither a < b nor b < a. Definition 1.^. A lattice is a set together with two binary operations A and V defined on the set and satisfying: (1) X y y is an element of the set (closure) (2) X y X X (idempotency)

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-5\>) U V y; z = X ^ (y ,j z) (associativity) (^) xjy=yjx (commutatlvlty) (5) X A y = X If ojid only if x V y = y (alternation) Convention 1.10. a < b if and only if a A b = a, LermR 1.1, A lattice is a poset under <, Proof . (1) a A a = a implies a < a, ( 2 ) a < b 8Jid b < a imply a = a A b = b , (3) a < b and b < c imply a A b = a and b A c = b imply aAc=(aAb) Vc=aA(bVc)=aAb=aso that a < o» Definition 1.6. A chain Is a subset of a lattice which is a simply ordered set under <. Definition 1.^ A metric space is a set together viith a real valued function (metric), ( , ), defined on the ordered pairs of elements of the set and satisfying: (1) (x,y) = 0 if and only if x = y (vanishing condition) (2) (x,y) + (z,y) > (x,z) (triajigle inequality) Lemma l.£. In a metric space, (3) (x,y) > 0, (non-negativity) On (x,y) = (y,x). (symmetry) Proof . (3) (a,b) = 1/2 ((a,b) + (.i,b)) > 1/2 (a, a) = 0 ih) (c,b) = (b,b) + (c,b) > (b,c) ajid symmetrioally implies (b,c) = (c,b).

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-6Deflnltlon 1,^. A metric lattice is a Ir.ttlce together with a real valued function (norm) , | | , defined on the set and satisfying: (6) X < y implies |x| < |y| (strong isotonicity) (7) |xi + |y| = Ix A y| + |x V yl (modularity) Corollary l.g. a (a V b) A c (semi-modular law). Proof, a < c implies a V c = a. Thus, a V (b A c) < (a V b) A (a V c) = (a V b) A o.

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-7Lemma 1,^. a < b and |al > |b| Imply a = b. Proof, a < b implies la| < lb| , a contradiotion. This implies a = b. Definition l.^., A lattice is said to be modular if (y A (x V z)) V (x A z) = (y V (x A z)) A (x V z). Corollary 1,^, A lattice is modular if and only if x < y implies (x V y) A z = x V (y A z) , Lemma l.^. Metric lattices are modular. Proof . |a V (b A c) | = |a| + |b A c| |a A b A c| = |a| + |b| + |c| ib V c| la V b V o| > |a| + |b| + |c| |a A b| |a V b V o| = |c| + |a V bj |a A b A o| = I (a V b) A c| and a < c imply a V (b A c) = (a V b) A c. Convention 1,11. (a,b) = |a V b| |a A b| , Lemma 1.10. A metric lattice Is a metric space under metric ( , ) . Proof . (1) .(a,b) = 0 if ajid only if |a V b] = |a A bl \Thlch is equivalent to a = a A (a V b) = a A (a A b) = b A (a A b) = b A (a V b) = b. (2) (a,b) + (c,b) = |a V b| [a A b| + |c V bj |o A b| = |a V b V c| + |(a V b) A (c V b)| [a A b A c| l(a A b) V (o A b)| = (|a V c| + |b| ! (a V c) A b| ) + (|a A c| |b| + |(a V c) A b| + ( | (a V b) A (o V b)| I (a A b) V (c A b) I ) > (a,c) since (a V b) A (c V b) > (a A c) V b > b > (a V c) A b > (a A b) V (b A c).

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Conventlon 1.12. B(a,b,c) If and only If (a,b) + (b,c) = (a,o). Deflnltloi^ l.ip. b Is said to be metrically between a and o If B[a,b,c] subsists, Lemma l.^l. B(a,b,c) If and only if (a V b) A (b V o) (a A b) V (b A o). Proof, (a V b) A (b V o) = (a A b) V (b A c) Implies (aAo)Vb=(aVo)Ab, The result follows from the proof of (2) in the previous lemma. Lemma 1.12. a (b V c,a A b) implying B(a A b,b,b V c), 2(a,b) + ((a V b,a) + (a V b,b) (a,b)) + ((a, a A b) + (b,a A b)) < ((a V b,a) + (a, a A b)) + ((a V b,b) + (b,a A b)) = 2(a,b) implying B(a,a J b,b). (a, a A b) = (a,b) « (b,a A b) = {(a V b,b) + (b,a A b)) (b,a A b) = (a V b,b), (a,b) = |a V b| ja A b| = (a V b,a A b) ,

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-9Gorollary lj±, B(a A b. ,a V b) and B(a,a ^. b,b) imply , a V A V (a, a y b) = (a ^ b,b). Corollary 1.^, The results of the previous lemma are valid in a metric lattice. Lemma l.lA. A necessary and sufficient condition that a lattice v;ith a real valued function, | | , on it, be a metric lattice is that it be a metric space under ( , ) vrhere (a,b) = |a V b| ia A b| . Proof . (1) a < b implies a < (a,b) + |a| = |b| . (2) |a| + |b| = (a V b,b) (a, a A b) + |a| + |b| = la V b| + |a A b| , Levma. 1.1^. A necessary and sufficient condition that a lattice vrlth a given metric ( , ) be a metric lattice vrith metric ( , ) is that B(x,xOyiy) B(x A y,y,y V z) hold. Proof. For some fixed element a, let |x| = (a V x,a) (a, a A x). |x V yi ix A y| = ((a V X V y,a) (a, a V (x A y) ) + (a, a A X A y) (a A (x V y),a) = (a V x V y,a V (x A y) ) + (a V (x A y),x A y) (a V x V y,x V y) = (x V y,x A y) = (x V y,x) + (x,x A y) = (x V y,x) + (x V y,y) = (x,y). One then applies the previous lemma, Le'Mna l.l6. In a metric lattice i;ith given metric, the norm | | is determined to within an additive constant. Proof. |xi = (x,x A a) (x A a, a) + |a|. Convention l.li. abc = (a A b) V (a A c) V (b A c) abc = (a V b) A (a V c) A (b V c) Si= (a J b) ; (a ; o) (b ; o).

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-10Leinma l.ll. abc < abc (serai -distributive law). Proof , a A b,a A o,b A c < a V b,a V o, b V o. Convention l.l^t. D(a,b,c) if and ozay If abc = abo. Remark 1,2, D(a,b,c) is symmetrlo in a, b, and o. Convention 1.1^. D( ) if and only if D(a,b,o) for all triples a, b, and c in the first parenthesis. Definition 1.11. A lattice is said to be distributive if D(x,y,z) is valid, Lenana I.IS, Distributive lattices are modular. Proof, a < c Implies a V (b A o) = (a A b) V a V (b A o) = abo = abo = (a V b) A c A (b V o) = (a V b) A o. Lemma 1,12, D(a,b,a) holds. Proof, (a A b) V (a A a) V (b A a) = a (a V b) A (a V a) A (b V a), Lernma 1,20, In a modular lattice a < o implies D(a,b,c), Proof, abc = (a A b) V a V (b A o) = a V (b A c) = (a V b) A c = (a V b) A o A (b V c) = abc, Lewaa 1,2X» In a modular lattice ((a A (b V c)) V A c)) J ((b A (a V c)) V (a A c)) = igc. Lemma 1.22. |abc| | (a A (b V c) ) V (b A o) | = |(a A (b V c)) V (b A c)| label. Proof, lib^l |(a A (b V c)) V (b A c) | = jb A (a V o)) V (a A o) | |abc| = \abE\ |c A (a V b)) V (a A b) | = I (a A (b V c)) V (b A c) I label , Corollary 1.6. I (a A (b V c) ) V (b A o) | = | (b A (a V o)) V (a A c) I ,

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-11 Convention I.I6. (p.,b,c) = |abol | abo I . Corollary 1.2. (a,b,o) > 0 and (a,b,o) = 0 if and only if D(a,b,c) holds, Lemma 1.21. (a,b,c) = -Cl/2)(|a J (b c) | 5 b) I (a J 0)1). Proof, (a J (b c)) )( ibi = (a J (b )( c)) (b ; c) and (a J (b )[ c)) J ibc = (a J b) )( (a J c) by direct computation. By modularity and the previous lemma, the result follows, LeSJSa 1.2^. In a modular lattice D(a,b,c) subsists if and only if (a A (b V c) ) V (b A c) = (b A (a V c) ) V (a A c) which is equivalent to (a J b) )( (a { o) = a J (b )( o) . Proof. Since a V b = a if and only if a A b = b, the same argument applies as in the metric lattice case. Convention l.jj. N(a) is equivalent to D(a,x,y). Definition 1.12. An element 'a' of a modular lattice is said to be neutral if N(a) holds. Convention 1.1^. N( ) if and only if N(a) for all a in the first parenthesis. Corollary l._^. A lattice is distributive if and only if N(x) subsists. Definition 1.11. a is said to be the relative complement of b with respect to c and d if a 0 b = c o d, Lemm 1.2^. a <> b = b O o implies a = c, if and only if D(a,b,c) is valid.

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-12Proof , abc = a ^ c. Leomg 1.26, A lattice is distributive if and only if relative complements are unique. Proof, If the lattice is not modular there exist a, b, and o such that a < c and a V (b A c) < (a V b) A o. Then b A ((a V b) A c) = b A c = b A (a V (b A c)) ejid b V ((a V (bAc))=bVa = bV((aVb) Ac), Othervfise, ((a A (b V c)) V (b A c)) J ((b A (a V c)) V (a A c)) = abc = ((b A (a V c)) V (a A c)) J ((c A (b V c) V (a A c)), Lemma 1.22, I^" D(a,b,c,d), then a A c = b A d and aVd = bVcif ejid only if a = b and c = d. Proof, D(a,b,c) implies a < (a V b) A (a V c) A (a V d) = abc abc = (a A b) V (b A o) V (b A d) < b. Similarly, b < a, c < d, and d < c. Definition l.lA. A lattice is said to be relatively complemented if x < y < z implies that y has a relative complement with respect to x and z. Definition 1.1^, A lattice is said to have a ^o^^^r flrnia upper bound or element, denoted by ^ , if ® x. Lemma l.^S. Lower -^^^^^ ^^.^ unique."" Upper Proof. I' I] J X implies ^<^'<^ implies ^ = *P' ({) ({)•-({) (1) = (|)« Definition l.l6. A lattice is s-id to be bounded if it contains upper and lower bounds. Convention 1.1^. The use of ® in a condition implies that the condition is valid only vrhen the lattice contains the respective bound.

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-13Deflnltlon 1.12. A relatively oomplemented, distributive, bounded lattice is called a Boolean algebra. Lenima l.£2.» ^ relatively complemented lattice xvith first element, the modularity postulate (6) for metric lattices may be replaced by (6') X A y = 0 implies |x| + |y| = |x V y| + |©|, Proof . Let c be the relative complement of a A b with respect to G and a. (|a| = |q| + |b| + |c|) + |b| = |a A b| + (Ic| + |bl) = |a A b| + |o| = |a V bj. Definition 1.1^. A set algebra on a given set is the set of subsets of that set under the set-theoretic operations. Corollary l.^,, A set algebra is a Boolean algebra under the operations of set intersection, set union, and relative set complement, • Convention l.?0, P( ) denotes the set algebra on the set in the parenthesis. Convention 1,21, [2^] denotes the set algebra on n elements. Lemma 1.30, [2^] forms a metric lattice under |{ }| = number of elements in { }, Proof. I I is, clearly, strongly Isotone and satisfies (6') of the previous lemma. Definition l.li. A set together viith a binary operation ( Jiixtaposition) defined for each ordered pair of the set is called a semigroup if

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(1) xy Is a member of the set (closure) (2) x(yz) = (xy)z (associativity) Definition l.?0. The subsemigroup generated by a subset of a semigroup is the set of all finite products of elements of the subset (elements may be repeated in the product) under the semigroup operation. Corollary 1.10. The subsemigroup generated by a subset of a semigroup is a semigroup. Definition 1.21. A group is a semigroup satisfying: (3) There exists an element ^ of the semigroup such that = X = (identity element) (^) For each x there exists x'"^ such that x'^^x = ^ ~ (inverse) Lemma l.^l* 1^ ^ group ^ and a""^ are imique. Proof , = 5%e» = jC*. a"^ = a"'^(aa"''"*) = (a~"''a)a"''' = a Doflnitlon 1.2?, A group is called Abelian if (5) xy = yx (comnutativity) Definition 1.22.. A subsemigroup of a group which is itself a group is called a subgroup of the group. Definition 1.2^, A subset of a lattice which is also a lattice under the operations of the parent Irttice is called a sublattice. Corollary 1.11, A subset of a lattice which is closed under the operations of the parent lattice forms a sublattice.

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-15Corollary 1.12, A lattice is a sublattice of Itself, Definition 1.2^. The set of elements, together vrlth the lattice operations, which are common to every sublattice containing a given set, Is called the sublattice generated by that set. Corollary 1.11, "^^e sublattice generated by a subset of a lattice Is a lattice containing the subset. Convention 1,22, L( ) denotes the sublattice generated by the elements appearing in the parenthesis, Lenima l.^S, The neutral elements of a lattice f ora a sublattice. Proof . N(a) and N(b) imply a'^b^(x^y)=a^ ; ^) I J y)) = J J ^)) f (a J (b J y)) = (la J b) J X) )[ ((a J b) J y) implies N(a J b) . Levma 1,12., The sublattice generated bya subset vrhose non-neutral elements generate a distributive sublattice is distributive, Proof. Let {a^} be the sublattice generated by the non neutral elements and {b } be the sublattice of neutral TT elements. Then {a., b , a^O "b } forms a distributive org TT sublattice, Lemraa 1.3^. A chain forms a distributive sublattice. Proof. a,b in the chain Implies a Ob = f or ^, b a a < b < c implies abc = b = abc.

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-16Corollary l.l^t. Any simply ordered set forms a distributive lattice when aAb = a, a
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-17Lema 1.^. u Is a stroncly Isotone fimction of x. Proof , b < 0 Implies (n,^ A b) V ^ < (a^ V o) 2. so that v((a^ A b) ^ — v((a^ Ac) V ^) ojid, hence, u(b) < u(c). b <: c and u(b) = u(c) imply Z (v(a A b) V a ) " ~ i 1 1-1 v((a^ A c) V a^ ^)) = 0, Thus, a^ A c < a^ A (c V a^ ^) = (a^^Ac) Va^^ = (a^Ab) Va^^
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Proof , Since the selection of the chain is arbitrary, every maximal chain has length > n + 1. Lemma l.'l-l. If b < < b and. c < ... < c are 0 n 0 n maximal chains such tliat b Ac = b = c and b V c = J k X X J b = c , then b^<,.,
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CHAPTER II PSEUDO-LIITEAR QUADRUPLES Convention 2.1. Q(a,l);c,d) if and only If B{a,c,b), B(a,d,b), B(c,a,d), ajid B(c,b,d) subsist. Corollary 2.1. Q[a,b;c,d] If and only If B[a,c,b], B[a,d,b], B[c,a,d], and B[c,b,d] subsist. Remark 2.1. Q(a,b;c,d), Q(b,a;o,d), Q(a,b;d,o), Q(b,a;d,c), Q(c,d;b,a), and Q(d,c;b,a) are equivalent. Lemma 2.1. Q(a,b;c,d) if and only if (a,b) = (c,d), {a,c) = (b,d), (a,d) = (b,c), and B(a,c,b) holds. Proof. By solving the betweenness equations. Lemma 2.2. Q(a,b;c,d) if ajid only if B(a,d,b), B(o,b,d), (a,c) = (b,d), and (a,b) = (b,c) are valid. Proof. Substituting the equalities we obtain B(a,o,d) andB(o,a,d). Lemma 2.^. QCa,b;c,d], Q[a,o;b,d], and Q[a,d;b,c] are mutually exclusive. Proof. Q[a,b;c,d] and Q[a,c;b,d] imply (a,c) + (b,c) (a,b) and (a,b) + (b,c) = (a,c) so that (b,c) = 0, a contradiction. The result follows symmetrically. Definition 2.1. a,b,c,d, in a metric space, are said to form a pseudo-linear quadruple if one of QCa,b;c,d], Q[a,c;b,d], or Q[a,d;b,c] holds (3, , -19-

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Corollary 2.2. If a,b,c,d form a pseudo-linear quadruple, then (a,b) = (c,d), (a,c) = (b,d), and (a,d) = (b,c). Theorem 2.1. ( (a,c) (b,d) )-'-/^ + ( (a,d) (b,c) )-'-/^ > 1/2 ~" ((a,b)(o,d)) with equality if and only if a = c, b = d, a = d, b = c, or Q(a,b;c,d) hold. Proof, Label so that (a,o) > (b,d), (a,d) > (b,o), and (a,c) (b,d) > (a,d) (b,o) by using the permutations (ab)(od), (ac)(bd), and (cd) respectively, leaving the original inequality unchanged. Then (r,b) < (b,d) + (r.,d), (c,d) < (b,d) + (b,c), (a,d) (b,c) < (a,c) (b,d), and 2(b,c)(b,d) < 2((a,o)(b,d)(a,d)(b,o))-'-/^. Accordingly, (a,b)(c,d) < ((b,d) + (a,d))((b,d) + (b,o)) = 2(b,c)(b,d) + ((a,d) (b,o))(b,d) + (b,d)^ + (a,d)(b,o) < 2(b,c)(b,d) + ((a,o) (b,d))(b,d) + (b,d)^ + (a,d)(b,c) < 2(b,c)(b,d) + (a,c)(b,d) + (a,d)(b,c) < 2((.'>,c)(b,d)(a,d)(b,c))-'-/^ + (a,o)(b,d) + {a,d)(b,c) < (((a,o)(b,d)P + ( (a,d) (b,c) )^^^) ^ equality and a ?^ o,d implying D(a,d,b), B(c,b,d), (a,c) = (b,d), and (n,d) = (b,o) which imply Q(p.,b;c,d) , By the previous corollary Q(a,b;c,d) yields the equality. Corollary. 2.^. | i(x,y)| | < o. Proof. Let X = ( (a,b) (c,d) )''/^, m = ( (a,c) (b,d) )^/^, and n = ( (a,d) (b.c) )^/^ l!(x,y)j| = x'^ + + n^ 2(Xm + Xn + ran) = (X m n) (m X n) (n X m) (X + m + n) < 0.

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-21Corollary g,k, A necessary and sufficient condition for four distinct points to form a pseudo-linear quadruple is that the determinant of their distances vanish. Lemma 2,^, (a,b) + (a,c) + (b,c) = 2(a V b V c,a A b A o) + (a,b,c) , Proof . (a,b) + (a,c) + (b,c) = | a V b | + | a V c | + |b V c| |a A b| |a A c| |b A o| = |a V b V cl + (|(a V b) A (a V o) | + |b V o| ) |a A b A o| ( | (a A b) V (a A c)| + |b A c|) = 2|a V b V c| + |abo| 2la A b A c| l abe l « 2(a V b V c,a A b A c) + (a,b,c). Lemma 2,^, B(a,b,c) holds if and only if a A o < b < a V o and D(a,b,o) subsists. Proof . aAc b O o = a O o and thus implies (a,c) = (a V b V c,a A b A c). The subsistence of B(a,b,c) yields 2(a V b V o,a V c) + 2(a A c,a A b A o) + (a,b,o) = 2((a V b V c,a A b A c) (a,c)) + (a,b,c) = 0 which implies D(a,b,o) and a b ^ c = Theorem 2.2. Q(a,b;c,d) is valid if and only if a O b = c <> d and D(a,b,c,d) holds. Proof, aOb = cOd if ajid only if a A b < c,d < a V b and oAd
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-22Proof . B(a,b,c) Implies (a J J[ (a J c) J[ (b J o) » (a J b) ; J c) = b J (a ; c) = b. (a ; b) (a ; c) (b y o) = b implies aAo
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-231 Lemma 2.10. Q(e V f,g V h,e V g,f V h). Proof, (e V f) V (g V h) = e V f V s V h = (e V g) V (f V h). (e V f) A (g V h) = (e A g) V (e A h) V (f A g) V (f A h) = (e A f) V (e A h) V (g A f) V (g A h) = (e V g) A (f V h). (e V f) V ((g V h) A (e V g)) = e V f V g V (h A e) = e V f V g = ((e V f) V (g V h)) A ((e V f) V (e V g) ) implies D(e V f,g V h,e V g) , and symmetrically. Lemma 2.11. Q[a,b;c,d] implies (1) a,b,c,d are pairv/ise incomparable (a and b are called incomparable if a b) , or (2) a < c and b > d (or a < d and b < c) and all other pairs are incomparable, or (3) a < c,d < b (or b < c,d < a; c < a,b < d;or d < a,b < c) and all other pairs are incomparable. Proof. Assume some pair is comparable. By proper labeling, say a < c or a < b, a < c implies c V (b A d) = (c V b) A (c V d) = (c V a V b) A (o V d) = c V d which with 0A(bAd)=cAd implies b A d = b so that d < b. a < b implies a = aAb = cAdrjidb = aVb = cVd. Thus, a < c,d < b so that a < c. Assume a < c,b > d and some other pair is comparable. By symmetry and transitivity we may assume tha.t a < b so that a < c,d < b. If in addition o < d (or d < c) then we have c=cAd=aAb=a, contrary to assumption.

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Lemma 2. 12 , (1') L(Q.(a,bjc,d) = {a A b,a,b,c,d,a A c,a A d,b A c, b A d,(a A c) V (b A d) ,a V c,a V d,b V c,b V d,(a V c) A (b V d),a V b}. Also, a < c and b > d Imply (2') L(Q(a,b;c,d) = {a A b,a,b,c,d,b A o,a V d,a V b} , and a < c,d < b imply (3') L(Q(a,b;c,d) = {a,b,c,d}. Proof . Observe tliat all of the elements are generated by a,b,c,d, (1') aAb=eAf=(eVfVg)A(eVfVh) A(eVg A (f V g V h) is the © of the set. a=eVf=(eVfVg)A(eVfVh). aVc=eVfVg. b=gVh=(eVgVh) A(fVgVh), aVd=eVfVh. c=eVg=(eVfVg) A(eVgVh). bVc=eVgVh. d=fVh=(eVfVh) A(fVgVh). bVd=fVgVh. a A c = e = (e V f V g) A (e V f V h) A (e V g V h), aAd=f=(eVfVg) A (eVgVh) A (fVgVh). b A o = g = (e V f V h) A (e V g V h) A (f V g V h), b A d = h = (e V f V h) A (e V g V h) A (f V g V h). (a A c) V (b A d) = e V h = (e V f V h) A (e V g V h). (a V c) A (b V d) = f V g = (€ V f V g) A (f V g V h). aVb=eVfVgVh=(eVfVg) V(eVfVh) is the 0 of the set. Prom the above representation in terms of e,f,g,h we see that the set is closed under Join (V), From the representation in terms of e V f V g,e V f V h,e V g V h, f V g V h, v/e see that the set is closed lander meet (A),

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-25Thus the set Is a sublattlce. (2') a < o fijid b > d implies aAc = a, aAd = aAb, bAd=d, (aAc)V(bAd)=aVd, aVc=c, bVc= a V b, b V d = b, (a V c) A (b V d) = c A b. (3') a < < b implies aAb = a, bAc = c, aVd = d In addition to the above. Note that (2') ajid (3') may be represented in terms of e,g,h and c:,h respectively. Corollary 2.7, D(L(Q{a,b;c,d) ) ) . Theorem 2,J.. The sublr.ttice generated by a pseudolinear quadruple is [2'*'], [2^], or [2^] corresponding to cases (1), (2), and (3)^ respectively, of Lemma 2,11, Proof. (1«), (2'), (3') have a realization P[e,f,g,h], P[e,C,h], P[g,h] satisfying Q[a,b;o,d] ejnA (1), (2), {3), respectively. The equality of any tvio elements in (1'), (2'), (3') implies a realization as a set algebra on fev;er elements and a further comparability on a,b,c,d. Remark 2,2, Consider the following tvjo modular lattices formed by (1) a < b < c < d. (2) e<(t)ajidajb = ajc = b5c=b5d = c;d = ®. In (1) D(a,b,c,d), a J b ?^ c J d, a 5 c ?f b J d, and a J d b In (2) neither D(a,b,c), D(8,b,d), D(b,c,d\ nor D(a,c,d) are valid. Accordingly, the tvro conditions a { b = c J d and D(a,b,c,d) are independent.

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-26Convention 2.3. The symbol, t, when used after "Theorem" "Lemma", et cetera, shall denote quantification for all a,b,c, d,e,f In a metric lattice. Theorem t 2,j^, A necessary and sufficient condition that DCa,b,c,d] imply a,b,c,d form a pseudo-linear qiiadruple is tliat the lattice be of the form {a^, ©,(()} where 6 7^ tt Q Im-Dlies a-O a = Z« 6 ^ TT ip Proof . By the previous remark, the condition Implies there exist no four element chains in the lattice. Hence, A 0 a * b Implies a y b = so that the lattice is of the required form. In any lattice of the form stated, D[a,b,c,d], by suitable labeling, yields a A b = c = © and a V b = d = ({), Remark 2..3.» Due to their simple structure, the multiple projective root lattices of the previous theorem have interest principally as sublattices. Definition t 2.2. A lattice in vjhich a<>b = c O (a V d) A (b V c) A If (a V b) A (c V d) > (a A c) V (b A d) .

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-27Inequalities 2 and k become equalities If and only If D(a,b,c,d) holds. Inequalities 1 ajid 3 become equalities if (a A b) V (c A d) < (a V b) A (c V d) and D(a,b,c,d) holds. 5 ((a V d) A b) V ((a V d) A c) Proof , (a V d) A (b V c) > y 5 ((b V c) A a) V ((b V c) A d) > (a A b) V (a A c) V (b A c) V (b A d) . Ineauallty 7 Is g an equality If ojid only If D(a,b,d) and D(b,c,d) hold. Equality In S Is equivalent to D(a,b,c) and D(b,c,d). Therefore, 7 and g are equalities if and only if D(a,b,c,d) subsists, vfhich in turn implies that 5 and 6 are equalities. Thus, i}is an equality if ajid only if D(a,b,c,d) is valid. Changing symbols, ((od), (AV), rjid (<>)), the same argument applies for 2, (a V b) A (c V d) > (a A b) V (c A d) ajid D(a,b,c,d) imply (a V d) A (b V c) A (a V b) A (c V d) > (a A o) V (b A d) V (a A b) V (a A d) = (a V d) A (b V c) which implies 3 is an equality, in vrhich case 1 is an equality. Corollary 2.10. Each of the following six expressions is equivalent to D(a,b,c,d). (a;o):a>) = (a>, ;(a;;a);,.^, ;), ,A V A, ,V A V A y A V (% , (° V 1) = o) , , 4) ^ (b ^ o) ^ (b ^ d) „ J ^' : = (a : o, ; (b « ; (0 ; a, Each of the following four expressions is equivalent to (a A b) V (c A d) < (a V b) A (c V d) and D(a,b,c,d). (a : 0) ; (b ; a) = (a I a) J (b o)

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-23(b ^ d) y (o ^ d). Remark t 2..^. a < o and b < d implying any one of the ten equations of the previous corollary Is equivalent to the modular law. In a modular lattice, a ^ b = c ^ d Implying any one of the ten equations Is equivalent to the distributive lav;. Theorem t 2,^, A necessary ond sufficient condition that a lattice be quasi-distributive Is that aO b = c O a,b,c,d distinct Imply one of the ten equations In the previous corollary. Theorem t £.6, Q(a,d;b,c) rjid Q(o,f;d,e) Imply Q(a,f;b,e) If and only if the lattice Is distributive, A , ^ A Proof , Assume dlstrlbutlvity, a ^ d = b ^ c and c J f = d ^ e imply d ^ (a 'J f ) = b 'J c f = d (b ^ e) and d )[ (a J f) = (d a) 5 (d J f) = (b c) J (d f) = A V , A ^, V A ^, V , A /a. A V . A V V A V A V A ^° V = V A V A ,,VA,VA,VA,\'^ / AV^AV ""^ A <=' V "> A ^' V A V A ^>> = V ^> A <^ V 1> A VAVAVAV VAVA <<^ A*' V V A V <^ a'^" = ''^ A^' V V (d^e) >,) = (a;d, ^(djf, ;

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-29(f J o) J (f 5 d)) = (b J c) (d J f) ((b ; d ); (c j d)) = (b y d) (c y f) = (d Nd y e) = d y (b (5 e) which AVA AVA AV Implies a f = b y e. Non-dlstrlbutivlty Implies that there exist a,b,c such that a ^ b = c ^ b with a c. Then Q[a,b;a A b,a V b] and Q[a V b,a A b;b,c] hold sjid Q(n,a A b; a A b,c) does not. Corollary t 2.11. If Q[a,d;b,c] and Q[c,f;d,e] yield distinct pseudo-linear quadruples, then Q[a,f;b,e] holds if and only if the lattice is distributive. Lemma t 2,Vi, Cl(a,b;c,d) and Q(c,d;e,f) imply Q(a,b;e, if and only if the lattice is distributive. Proof . Non-distributlvity implies the existence of a,b,o such that a J b = c J b which implies thr.t Q[a,b;a A b, a V b] and Ci[a A b,a V b;b,c] hold while Q(a,b;b,c) does not. Corollary t 2.12. Ci[a,b;c,d] andQ[o,d;e,f ] imply Q[a,b;e,f] if and only if the lattice is distributive. Corollary t 2.11. QCa,b;c,d], Q[c,d;e,f] end a,b e,f imply Q[a,b;e,f] if cjid only if the lattice is quasi-distributive. Corollary t 2.1^. Q(a,b;c,d) pjid Q(a,b;c,e) imply d = e if raid only if the lattice is distributive. Convention 2 A, In the following theorem a,b vrill denote the ordered pair of elements a,b.

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-30Theorem 2.2. The relation, a,bn^c,d if aiid only if a<>b = cOd, Is an equivalence relation in the set of pairs of elements of a lattice, Moreover, a,b'>-'C,d If and only if Q(a,bjc,d) , a,b ^c,d if and only ifaOd = bOc, a,b'\^c,d if and only If Q(a,d;b,c) are equivalence relations if r-jid only if the lattice is distributive. Proof . For all of the relations: ajb'vajb and, a,b^c,d implies c,d^a,b. Transitivity follows from previous theorems and lemmas. Theorem 2.g. aOb = c <> d implies (a,b,c) + (a,b,d) = (a,c,d) + (b,c,d) , Proof . aOb = cOd implies (a,c) + (b,c) = (a,b) + (a,b,c), (a,d) + {b,d) = (a,b) + (a,b,d) , (a,c) + (a,d) = (c,d) + (a,c,d), 8Jid (b,c) + (b,d) = (c,d) + (b,o,d) imply (a,b,o) + (a,b,d) = (a,c,d) + (b,c,d). Corollary 2,1^, aO b = cO d implies that D(a,b,o) and D(a,b,d) are together equivp.lent to D(a,c,d) and D(b,c,d). Corollary 2.l6. a O b = c O d together vrith any three of D(a,b,c), D(a,b,d), D(a,c,d), D(b,c,d) imply the fourth of these. Corollary 2 .12. a <> b = c O b = c O d and j?jny one of a < b, a>b, cd imply D(a,b,c,d),

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-31Corollary 2,.12.» a <> b = c O
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-32Corollary £.50, a J b = c J d implies D(a J b,a J c, a y djb y c,b y d . nbo . abd. acd . bed ) . Proof, Similar to that of the previous lemma. Theorem abc J ibd = i^d J bid, ( (a J c) )[ (b J c) ) J ^ ^^ V /, A ... _ A V , A A A V „ A ((a ; d) ; (b ; d)) = ((a ; c) ]( (a J d)) J ((b J c) ^ (b J d)), a^d D(a 5 b,a 5 c,a J d,b J c,(a J c) )( (b J c),(a J d) )[ I ^^'^^ I I !1 ^^(^ !1 ^) ! ^ d)) Wies a ^ b = V VAV VAV V c y dj and conversely. Proof. ajb=ajbj abc J abd = a J b J Hod J bed = (a 0 b 5 0) ,Na Jb Jd) = ((a ; c) (a Jd)) I ((b J c) (b 5 d)) = ((a J c) (b J c)) ; ((a j d) (b ; d)) = (a ; c j d) ;[(b jcjd)=cjdjibijibd=o0djiidjbcd = cjd. Definition g.±, A "forbidden picture" for ouasl-distributivity is a modular lattice generated by four distinct elements a,b,c,d such tli.nt aO^ = cOd but Q[a,b;c,d] fails. Corollary 2,?1, Forbidden pictures are not distributive. Remark 2.4, The simplest forbidden picture is the multiple projective root lattice (2) of Remark 2.2; the most coraplic-ted, the free modular lattice on aO b = c O d (2),

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-33Porbldden pictures may be classified by the number of nondistributive triples (nbc < abc) of a,b,c,d; that is; 2,3, or They may also be classified by the number of pairings ; a<>b = cOd, aOc = bOd, aOd = bOc which subsist, More theji one pairing of distinct elements always implies non-distributivity and the subsistence of three pairings characterizes the multiple projective root lattice. Lemma 2,1^, No lattice equation in three variables is equiv,'?.lent to quasi-distributivity. Proof . In the multiple projective root forbidden picture, every sublattice generated by three elements is qua s i -di s t r ibu t i ve , Remark 2,^, Kore can be said for relatively complemented modular lattices (6, 1^, 27) as follows: Lemma 2.12., Ln. a relatively complemented distributive lattice B[a,b,c] holding implies the existence of d so that Q[a,c;b,d] is valid. Proof . Let d be the relative complement of b vfith respect to a and c. Then dO b = a O c ond. h ji g, a 7^ c, and a 7^ b which imply d 7^ a, d 7^ b, and 6. ^ o respectively. Lemma 2,20. In a relatively complemented, quasidistributive metric lattice, a J b (a J b) (a J c) , (a J b) V A (b y c) implies D(a,b,(c A (a V b) ) V (a A b) ) , Proof. Let c' = (c A (a V b)) V (a A b). There exists d such that dOc'=aOt), d = c' implies a = b, and d = a or b implies a contradiction of hypothesis. Hence, if

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-3^ a,b,c' are distinct, then D(a,b,c\d) follovrs. Remark 2»6_, The previous lemma holds for non-oomplemented lattices V7hen the required complement d exists. The lemma, vrtien applicable, and uniqueness of pairings of distinct elements (that Is, aOb = cOd and a,b,c,d distinct imply the falsity ofajc=bjdandajd=bjo) characterize quasl-dlstrlbutlvlty. Remark 2,X» Relatively complemented, q\ia si -distributive lattices are, in general, distributive. Exceptions are special cases such as the prime projective root lattice (a ; b = a ; c = b J c = ^; 0 < ({)). Theorem 3. 10 . A lattice which is also a metric space is a metric lattice If and only if Q(x,y;x A y,x V y) and B(x A y,y,y V z) hold. For relatively complemented lattices the second condition may be omitted. Proof . Q(a,b;a A b,a V b) Implies B(a,a J b,b) and B(a A b,a,a V b). In a relatively complemented lattice, given b,c,d there exists 'a' such that a A b = b A c and a V b = b V d; hence B(b A c,b,b V d) ,

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CHAPTER III THE TRAIISPORriATION u(x) = |x| + r(a,x) Leimna 1.1. a < b and o < d Imply (a,d) + (b,c) > (a,c) + (b,d) with equality holding if PJid only if a V c > b V d. Proof , (a V d) A (b V o) = a V (d A (b V c)) = a V ((d A b) V c) = (a V c) V (b A d) > (a V c) = (a V c) A (b V d). (a A d) V (b A c) = b A (o V (a A d)) = b A ((c V A d) = (b A d) V (a A o) < b A d = (b A d) V (a A c). Thus la J d| + lb J ci = |(a J d) y (b c)| + la S S Nl l(a V A V + |a V b J c J d| = |a J c| + |b J d|, k n Lemma 1,2, IfEsE s >0 for 0 < k < n, i=0 i i=k+l i 0 < r > |r I , and i < J imT)lies r > r , then Z s r > 0. ^ ^ i J i 1 i Proof, n = 0 implies s^>Ososr >0, n=l 0 0 0 implies s > I s I so that sr +sr >sr +|srl> 0 1 00 11-00 1 1|b I )r > 0. Assime the lemma for 1 < n < m. We may 0 1 0 ~ also assume that 0 for all i. m > 2 implies there exists j such that s and s have the same sign. Hence, J+1 <(sr+s r )(s+B )~ 0, i=J+2 i i -35-

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-36Theorem ±,1, A necessary and sufficient condition that u(x) = q|x| + r + Z s (a ,x) ; © < a < ... < a < d), ill 1 m n be a strongly isotone fimction of x Is thct q + E s > n 1=1 1 Z 8 ; 0 < m < n, l=m+l 1 Proof . Select some element a > a , u(x) stronerlv n+1 ~ n isotone implies 0|a|)+ m tiH-1 m m+1' m 2 8 (da I |a I) (|a I ia I)) + 1=1 1 m+1 1 mi ^ m n Z 8 (da I |a J) (|a I |a I)) = (q + Z s Z s) l=m+l i 1 m+1 1 m 1=1 i i=mfl 1 (I a I I a i ) so that q+Zs > Z s;0 0, and J < k implies a < a so that (a ,d) (a .c) > J ~ k y J (a ,d) (a ,c). Therefore u(c) = u(d) (q(c,d) + Z s ((a .d) k ill (a^,c)))< u(d). Corollary ^.l. u(x) = q(x) + Z s (a ,x): 0 < a < . 1111 ... < a < q), is strongly isotone if and only if u is n strongly isotone on a |r| . Corollary u(x) = q|x| + r is strongly isotone If and only if q > 0, Convention I.l.(^;a,b) = u(a) + u(b) u(a A b) u(a V b). Corollary u is a modular functional if and only if (u;x,v) = 0.

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-37Conventlon ^,2, N(s ;a ) if and only if Z s (a ,x,y) = 0, 11 1^1 Definition ^.1. A set of coefficients s^ and lattice elements a Is said to be a pseudo-neutral set If N(s ja ) holds, 1 11 Corollary l.'S. N(a ), for all 1, Imnlles N(s ;a ), 1 1 1 Corollary ±,6, N(l;a) holds If and only If N(a) subsists. Lemma u(x) = qjxj + r + Z s (a ,x) Implies 111 (u;b,c) = E s (a ,b,c) , 111 Proof . u(b) + u(c) u(b A c) = q( |b| + Ic| [b A c| |b V c|) + E s ((a ,b) + (a ,c) (a ,b A c) (a ,b V c) ) = 1 i 1 1 1 1 E s ((|a V b| + |a V c] [a V b V o| |a V (b A c) I ) + ill 1 1 1 (|a A (b V c)i + la A b A o| ja A b| |a A o|)) = 1 1 11 2 B ((!(a V b) A (a V c)l |a V (b A c)|) + (la A 111 1 1 1 (b V c)| |(a A b) V (a A c)|))= E s (a ,b,c). 1 1 ill Theorem ±,2, u(x) = q|x| + r + E s (a ,x) Is a 111 modular fimctlonal of x If and only if M(s ;a ) Is valid, 1 1 Proof . By the previous lemmas. Corollary J^.J. u(x) = q|x| + r + s(a,x) is a modular functional of x if njid only if N(a) subsists. Corollary u(x) = q|x| + r is a modular functional of X, Remark There exist pseudo-neutral sets containing non-neutral lattice elements. For example; in the free modular (metric) lattice generated by a,b,c (1), a and (a V (b A c) ) A (b V c) are distinct non-neutral elements

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-32vrhlle (a,x) ( (a V (b A c) ) A (b V c),x) is a modulp.r functional of x. Remark ^,2, Consider the folloiJing problem. Given {r ,a } , In a metric lattice, v/here a < a Implies r < r , ^ 1 J ~ k j ~ k does there exist a norm u on the lattice such that u(a ) = r^ for all 1? This vjlll be referred to as the n-element problem. If the r are required to be sufficiently close to 1 the corresr>ondlng | a | , the problem will be called the local 1 n-element problem. Convention In this cliapter a a b If and only If u(a) < u(b) for all lattice norms u. Corollary 1.2.. nc Is a quasl-order relation. Lemma a < b Implies a a b. Proof , a < b Implies u(a) < u(b) and this Implies a nc b. Theorem ^.3,. If N(a) o^r N(b) subsists, then a a b is equivalent to a < b. Proof, a < b Implies a a b, a < b Implies Ia| < |b| so that b a a Is false, a b Implies |a A b| < |a|,|b! < |a V b| so that |(|a| |b|)l < [a V b| |a A bl = (a,b) and |(|a| |b| ^ s)| < (r.,b), where s = l/2((a,b) ia| > 0, 11(a) holding Implies that u(x) = |x| + ([aj |b| + s)(a,b) (a,x) Is a norm with u(a) = a pnd u(b) = |a| i s. Then u(a) > u(b) so that neither a a b nor b a a.

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-39Theorem i^iL* ^ necessary and sufficient condition that a be a partial order relation on a lattice is that the lattice be distributive. Proof . Sufficiency follows from the previous theorem. If the lattice is not distributive there exist r. ,b,c v;ith (a,b,c) 0 which implies u((a A (b V c)) V (b A c)) = u((b A (a V c)) V (a A c)) and (a A (b V c)) V (b A c) ?^ (b A (a V c)) V (a A c). Hence, a is not a partial order relation. Corollary 1.10. A necessary and sufficient condition tliat £c coincide vrlth < is tlir.t the lattice be distributive. Lemma 1,^, The two element problem is solvable < if a > b. Proof. The problem is trivial for a = b. a ^ b and r J s imply (r s) ( | a| | b| )~''" > 0 and, hence, u(x) = (r s)(|a| ibi)"^ix| + (s|aj r|b|)(lal |bi)"^ is a norm with u(a) = r rmd u(b) = s. Theorem ^.5^, A necessary and sufficient condition that the two elenent problem be solvable for a,b is that a a b if and only If a < b, pnd b a a if ajad only if b < a, < Proof . If a ~ b the problem is solvable, a b and u(a) = r J s = u(b) imply the falsity of ^ a ^. Corollary 1.11. N(a) holding implies thp.t the two element problem is solvable for a,b.

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r -J+oCorollary 2.1£. The tvro element problem Is solvable for all pairs of elements In a Ir.ttlce if pjid only If the lattice Is distributive. Remark ^.J,. The quasl-order relation a may become quite complex compared vrith the partial order relation <. Examples show that one may have a jc b in a lattice, but not in any finite or proper sublattice. Convention 3.,^, u is the normtrans format ion sending |x| into u(x) and uv is the transformation sending |x| into the trajis format ion product u(v(x)). Corollary l.li. (uv)w = u(vw). Proof, (uv)vt(x) =u(v(w(x))) =u(vw(x)). Corollary j,.!^. u(x) = |x| is the identity transformation. Convention G is the transformation semigroup generated by all transformations u of the type u(x) = |x| + r(a,x), where u is a strongly isotone function of x ajid r is a function of scalars and norms, | 1 , of fixed lattice elements in a bounded metric lattice. Remark 1.^, The boundedness of the lattice in the previous convention is used only to extend the generating elements to those of the form u(x) = q j x| + r + s(a,x) . JSeraark LS. u(x) = |x| + r(a,x) and v(x) = |x| + s(b,x) commute to t^ithin a constajit function. Since uv(x) uv(x) = rs(|a| |bi), u and v commute if and only if r = 0, B = 0, or |ai = |b| .

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Leanin ^,6. u(x) = q I x| j q > 0 is In G. Proof. For q | 1 let b = J ojid c = ^. Then t|c| > i|b| > lq|b| so tha.t s = (|c| q|b|)(|c| |b| )''• > 0, which implies v(x) = |x| 1 (s l)(c,x) = s|x| + (1 s)ic| Is in 0. Similarly w(x) =(q/s)lx| + (1 q/s)|b| is in G. Hence u(x) = yi;{x) = sw(x) + (1 s)w(c) = q|x| + (s q)|b| + q(l s)s-^|c| + ( 1 s)(l q/s)|b| = qjxl + (s q)s"^|b| + q(l s)s"-^|c| = q|x| is m G. Leng^a ]^,2, u(x) = |x| + r is in 0, Proof. Let b = either © or (j) so that |b| ji 0. There exists n > 0 such that |m"-^| < |b| which implies s = (1 + r(n|b|) ) > 0. From the previous lemma and its proof, v(x) = s"^(s|x| + (1 s)ib|) = |x| + (1 s)s"-^|b| = |x| + r/n is in G. Hence, u(x) = v^(x) = |x| + n(r/n) = |x| + r is in G. Corollary. 1.1^. u(x) = q| xj + r; q > 0 is in G. Lernma i.g. If u(x) = |x| + r(a,x) is in G, then u(x) has an Inverse in G. Proof. If a = ^ then 1 1 r > 0 implies v(x) = (1 1 r)""^ (|x| t r|a|) is in G. In the contrary cr.se, |r| < 1 implies v(x) = (1 r2)-^|x| r^lal r(a,x)) is in G. In either case, by direct computation, uv(x) = |x| = vu(x) . Lessja i.^. u(x) in G implies u(x) has an inverse. Pro^. u is a finite product of generators each having an inverse. The product of these inverses in reverse order is the required inverse of u.

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Lemma 1.10. If u(x) = qUl + r + E B^(n^,x) ; © < a. ^ < . . . < < ({), Is strongly Isotone, then u(x) is In G, Proof . The lemma is valid for n = 0; assume the k n result for n=m-l.q+Es > Z s;0 Z s;0 1 s 1 . n i=l 1 i=k+l i " "" 1=1 i n n-1 Therefore v(x) = (q s ) 1 x| + Z s (a ,x) Pnd w(x) = (q + n 1=1 1 1 n-1 Z s )lx| + s (a ,x) are in G, By direct computation, vtt(x) = i=l i n n n-1 n n-1 (q + Z s s )(q|xl + Z s (a ,x)) s la 1 Z s vxhich implies i=l i n i=l i i n n i=l 1 * u(x) = q|x| + r + Z s (a ,x) is in G, ill Lemma 3.11. u(x) = q|x| + r + Z s (a ,x) and N(s ;a ) ill i 1 holding imply the existence of q',r',m,t ,13 ; 1 < J < m, m J j such that u(x) = q' U^! + I*' + ^ t (b ,x) ; © < b < ... < b < (j) J=l J J 1 ^ and N(a ) for all i implies II(b ) ; 1 < J < m. Proof . This is a corollary to Theorem 4,2. Lemma 1.12. All elements of G may be put into the form u(x) = q|x| + r + Z s (a .x). i i ^• Proof . The generating elements u(x) = \x\ + t(b,x) are of the desired form. Assume the products of k generating elements are of the form v(x) =q|xl + r+ Z s (a ,x). By i i i direct computation uv(x) = (q t Z s )|x! + (r t Z s 11 11 ((a ,b) 2!a I |bl) + Z s (a ,x) + t(b ,x) + t Z s ( (a V .1 i ill ill b, x) (a^ A b,x)) wiiich may be expressed in the required form.

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Corollary J..I6. All modular elements of G may be put in the form u(x)=q|x|+r+2:s(a,x);©

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m Lemma 1.1^. 2u(x) = S ((v(h ) v(b ))/(lb I 1=1 1 1-1 1 ib. J) (v(b ,) v(b ))/(lb I !b |))(b ,x) + ((v(b ) i+J1 1+1 i 1 1 v(b ))/(|b I |b 1) + (v(b ) v(b ))/(|b ! |b |))|x| 0 10 m+1 m nH-1 m v(b )(1 + (v(b ) v(b ))/(|b I |b I)) v(b )(1 + 0 1 0 1 0 ra+1 (v(b ) v(b ))/(!b I lb 1)^ where 0 = b^ < . . . < b m+1 m ra+1 ra o n+1 and V Is a metric lattice norm, Implies that u Is a strongly Isotone function of x ajid u(b ) = v(b ); 0 < j < m+l. Proof . By direct computation and the fact that u Is strongly Isotone on b^ < , . , < b 0 m+1 Theorem ^.7., If u Is a norm on a metric lattice vdth finite maximal chain, then u corresponds to a transformation in Ct sending I | Into u. Proof . u(x) of the previous lemma is such a transformation where b .<...< b is the given maximal chain. 0 ra+1 Theorem ^.S. If v is a norm on a metric lattice and N(a^) holds for all 1, then there exists a norm u corresponding to a transformation In G such that u(a^) = v(a^) for all 1, Proof . u(x) of the previous lemma is such a transformation where b < ... < b is the chain of neutral elements ^ ra deternnlned by {a }.

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CHAPTER IV THE DETERMINANT I I (a, ,a ) I 1^ Convention ^.1. i* = 1 + characteristic log^i; that is, i» is the number of di{^its preceeding the decima.l point in the binary ex' mansion of i. Corollrry ^.1. i = J» if and only if 2^""^ < J < 2^, Convention if. 2, Define a"'" = a , a^''= a^^ A a*'. Corollary ^.2, Given {a^} then {a^} and {a*^} are uniquely defined. Corollary ^,3, a^"^ < a'^.a^ < a^^"*""^. Lemma 4.1, In a metric lat'-.ice; N(a ), for all 1, 1 ^ Implies N(a^) for all i,j. Proof , The neutral elements form a sublattice. Convention 1 a j if and only if 1 + 1/2 < i*-i» (J + 1/2)2^ ^ , Corollary ^.U. If i» = J», then 1 a j if and only if i < J. Lemma |£.£, {a, 1} is a simply ordered set. Proof , i + 1/2 < (1 + 1/2)2^*"^* implies i a i. 1 a j and J a k imply 1 + 1/2 < ( J + 1/2) 2^*"" (k + 1/2) 2 2 < (k + 1/2)2 " , and this Implies 1 a k.

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1 a J nnd j a i Imply 1 + 1/3 < ( J + l/2)2^*"-^*< (i + 1/2) 2 2 =1 + 1/2 v/hich, in turn, implies i + 1/2 = (J + 1/2) 2^*""^* and J + 1/2 = (i + 1/2) 2"^*"^*. Since i and J are inte£;:ers , i* = J* vrhich implies i = j. Either i ii J or i + 1/2 > (j + 1/2)2^*"^* vrhich imi)lies J + 1/2 < (i + 1/2) 2^^ vrhence j a i. Convention i j if rnd only if 1 a J and i 7^ J . Lemma ^.3., 2^ < J < k < 2'"" IraplleG a^ — Pro-^f . For n = 1 the lemma is vacuous. If n = 2, then 2 2, JJ — h h ~" n n+1 n-1 U even) If 2 < J < k < 2 , then either 2 < < (J l)/2 (J odd.) k/2 (k even) n < 2 or j is even and k = ,1 + 1. T' ese iranly (k l)/2 (k odd) j a^^ A a (J even) a^ A a (k even) '1 ^ a;j-l>/2 V a'^-^'/^ (3 odd) a(=-l>/2 V aC-^/^ (K odd) i h = a^ or a^ = a^^^ A a^^^ ,< p.^^^ V a^/^ = ajj, respectively. 1 k Hence, a'f < a, . 1 h Corollar y h_,^, 2^''^ < J < k < 2^ implies a^ < a^^ I.emma kA, n > k* > 1 imolles 2^"-'< (2k + 1 ) 2"""'^*~'^< 2^. Proof. 2^-1 < 2^^-l(k2^^*-^ + 2-^^*) = (2k + 1)2^-^'^*-^ = 2^(k + 1/2)2"^'* < 2^. Lerrna n > k^^ > 1 r>nd 2^"-'< 1 < (2k + 1 ) 2^''^'*~''" < J < 2^ imply a^ < a^^ < a^.

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Proof . The lemma is vacuous for k* = 1, n = k* + 1 > 2 Implies 2^'^"-' k* > 1, p < i < h — ' — n-k* n+l v> T (i even) (k + 1/2)2'' < J < 2""^^ implies 2'^"^ < < (i l)/2 (i odd) (2k + 1)2 < < 2 which implies a^ (J l)/2 (j odd) * h \^ ^ ^^^^ (1 even) k ^ ^^^"^ (J even) ^(i-l)/2 ^ ^(i-l)/2 ^^^j < a < (j.i)/2 ^ ^^^^ h h Lemma ^£,6 , i , J > 0 and 1 a J imply a^ < a'' » Proof , i a J PJid i» = imply 2^*"-'< i < j < 2^* which Implies a < a*^, i a j rjid i« < j* imply 2 < J + 1/2 < (i + 1/2) 2''*"^''' which implies 2.^'^''^ < J < (21 + 1) 2*' so that a-^ < a*^. i a J and i* > J* imply (j + 1/2) 2 < i + 1/2 < 2 vrhich implies (2j + 1)2 ^ < 1 < 2 so that a^ < a"' , Corollary ^,6, {a^} fomn a chain. Convention U^,^, Given {s, }, define s*^ = s"^ Z s , 1 k=J*+i k Corollr.ry ^£,2. 1 £ 2"'""'' < J,k < 2^ implies s"^ = s^^ Thooren ^.1. u(x) = Z s |a ^ xl ojid Z s (a .o'^.b) = 0 n, 1 1 1 V i i i 1 < J < 2 , l..ply u(b) =^r^ ^ b| +'^2/s,U^ J b|; 0 < m < n. m 0. ^ J^.iS,(a^,a'',b) =^f|;^iS^(a^ ,a^b) + ni
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™ m m ni
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rn J A m
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-50Lemmg. 4.^. k* = ((3/2)2^^* (1 + k))*. Proof . 2^*-^ < (2^*-^1) + (2^* k) = (3/2)2^* k* k»-1 k* (1 + k) = 2'' 1 (k 2 ) < 2^ . Corollary (];) = k* = {k)». Corollary U.ll^. 3^=3^= s^. Corollary (^) = ( k_) = Jc and (F) = (T) = ^. Lemma ii^.lO, k a 1 a k , Proof, k < l/2( (3/2)2^^* 1) < k so tha.t k « 1/2 < k* (3/2)2 < k + 1/2. Thus, Ic 2 1 2 Corollary ii.l6. ]£ = Ic if and only if k = 1, Corollary ^.IZ. a^ < a^ < a^. Lemma it'll* k a A if and only if a F. Proof , k a A if ajid only if k + 1/2 < (X + 1/2)2^^*"^* But this is equivalent to (3/2)2^* (1 + k) + 1/2 < ((3/2)2^* (1 + X) + 1/2)2^ which is equivalent to -k 1/2 < . , ^ k*-X* ~ (-X 1/2)2 which holds if and only if X a IF, Corollary Either < a< a^ < a^^ or a^ < a< FX-" " a < a , Convention '±.^. The prime in 2* will be used to denote k k a 1 in the summation; that is, k = k and k 1, Convention i.^. = r^(, Z ^s^ + s^) + s^( E + r^) . TMorem k^.J. z r^(a^ ,a^ ,a^^) = 0 for all J,k implies ifjVi(-i»-j) = r^^s(-^-^)Proof. Note that (a^.a^'^a^) = 0. ^^j^^s^ (a^ ,a^ ) =

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-51(a^a^ + (a^a'^)) + ItV( (a'^.a^) + (a'^.a^)) + E'rV((a\a^ _ It X ' (al.aN) + rlsl(a\al) =1rV2(a\a^) + E' A^2(a\a'^) + r'rV((a\a'^) + (a'^.a'^)) + Z-rV(a\a'^) + ST ^ sNa\a'^) ^ X ^g; (A' Hr^^2(a^a'^) + E-rV^2(a^a^) + |.(rV + rV) (a^a^) = |'(2A^s^ + 2sV^/ + s/s'^ + r'V + r^'')(a\a^) K. ka\ kaX K kaXak kaXaF k ?s Corollarjr If il(a^) holds for all 1, then Legima ii.l2. i < 2 Implies 2 a 1 a 2'^. Prioof. i» < J implies (2*'"''' + 1/2)2^*"'' = 2^*""-'+ (1/2)2^*-^ < 2^*-l + 1/2 < i + 1/2 < 2^* 1/2 < 2^* . (1/2) 2^*J < (2'5 1 + 1/2) 2^*-5. Lemma h,!^. 2^"^ = 2"" 1 and 2^^ . 1 = 2"^"^. Proof. (2^-1)* = (2^ D* = n and 2''"'^ + 2'' 1 = (3/2)2^ 1. Convention ^.10. For j a 1 let J • denote the succesBor of j in the simply ordered set {a,i|l < i < 2^}; that is, J a j' and j a k imply j» a k.

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-52> 0. Corollary ^.20, j' e:clsts and is unique. Lemma ^.l4. s = 0 Implies Z =» -22 s^. 7" 0 _4fxiaJ incj. Proof , si = = implies Z si = E si, iili Tai Lemma i.l^. s = 0 ajid J a 1 irax)ly -2Z = ( E s^)^ Proof , j = 2^"-^ Implies -2Z = -2^^ = -^s'^( E + ) = -i^-s''(E s, s^) = (2sJ)2 = (E 28'')^ = ( E i i Jalaj Assume the lemma for J a 1. -2E = -2E 2t''* = Ixy ss i2_j ss ss ( E .s^)2 -lfsJ'( E s^+s^') = (( E s^ + 2s^')2 _l^s^' Jf^i^J yrilay mctf (E s^ + 2s^') + (2s'5')^'^) = ( E s^)^. i r-xiajConvention ^.11. ^ in an equation denotes = and tliat the equation is latticeor number-theoretic and is consequently invariant under clian^e of metrics pjid norms. Theorem E r. r (a, ,a ) -2r E r-(a ,a ) =0, 1 j i' j 01 i 1' n' ' \ ^ \' ^'^^ f ^l^^i'^J*^'^^ " ^O^^'^^j'^^^^ ^ ° '^^^ imi^ly E r . |a 1 ^0, ill Proof. Define s = r ; 1 < i < n, and s = r r , i i n n 0 Then = 2 = 2 r^ = 0 mid E s^(a^,a^ ,a^') = 0. E'( E s^^)2((a\a'^') + (aV^')) =-2E'( E t" )((a\a^') + A ^^aka^ A IzaX ss (a\a^)) = -sz t'^ r.((a\a'') . (a^,a^)) = -sft'^ (a\/) k ssij^ Ic ss' ' '

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-53-2Z s s (a ,a ) = -2( E r r (a ,a ) + E r (r r )(a .a ) + l,j i J i J i^^jin iji j l??nin 0 i'n Z (r r )r (a ,a ) + (r ,a )) = -2( E r.r (a. ,a ) n 0 n n j n 0 n n i j 1 j I'j 2r Z r (a ,a )) =0 imT:>lies either Z _s' = 0, or a = a Ol i i n XakaX X rv(a^,ai) ic (a\a'^') ajid a = a ; for X a 1. Thus, Z's^^ ^ Z's Z' ^ n ^ t k (a^Sa^) k k:iX (a^,a^ ' '1 k^a »^ ) . _ _ k^''^ >^ ^ ^ Z'Z' s r T' -1/2Z« Z s^^ ^ r-, 0, Note that t X kaX {a'^,a'' ) X X:tkaX (a^,a^ ) (a. .a'^ja ) = 0. Consequently, r (a ,a ) = Z r (a ,a ) Ai oni^ijL]_ Z s (a ,a ) = Z s. (a^,a J = Z s"^(a-^a-'-) = Z's''((a^^aJ + IJ-iJ-i-i-llk k 1 (a^,a )) = 0 so that r = 0 v/hich implies Z r la I ^ Z s^la^l t -L 0 1 1 i' ' ' Z s^'la I ^ Z's^'((a^',a ) (a'^a )) ^ 0. t k 1 k 1 1 Corollary 4.21. If, in addition, Z rj^(a^ ja"' ,b) = 0 for all j, then Z r (a, ,b) ^ 0, i 1 ^ Proof . Usinf; ^ V ^ t-marked lines in the previous r)roof, vre pret Z r |a b| 5i o. i i' i V ' Leana 4.16. Z r^|a^| 9^ 0 ejid N(b) holding imply Z r^(a^,b) ^ 0. Proof, u(x) = 2lxi + (b,x) being a norm implies Z r^(a^,b) 9i s r^laj + Z r^(a^,b) Si e u(a^) ^ 0.

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Corollary ^.22. Z r |a 1 0 and N(a ) holding for 111 1 all i imply Z r (a, ,x) ^ 0, i i i Proof . As L({a },b) is distributive, Z r (a ,b) ^ 0 X 1 i in the sublattice; which implies Z r (a ,x) S! o in the lattice. i i i Theorem If n > 2 njid il(a^) holds for all i; then ll(a -a )|| = 0 is equivalent to the existence of {r }, ^ J i|J i not all 0, such that Z rja^l ^ 0, Proof . I 1 (a ,a ) I 1 = 0 if and only if there exist ^ J 1,J {r }, not all 0, such that Z r, (a, ,aj =0 for all l. But the i i 1 1 J latter is valid if and only if Z r la I ^ 0, r not all 0. i i i i * or a = a v^hich is equivalent to Z r la 1 Si o. r not all 0. In 1 i i i ' or |a I |a i = 0. n Corollary 4.2J_. | i (a^.a^) j I 0, m > n > 2, and IT(a ); 1 < i < n,imply | | (a ,a ) \ {"^ ^ ^Si o. i m j^J=l Definition ^,1, A subset of a metric space is said to be equilateral if all distr noes botvreen distinct points of the subset are equal. Convention ^.12. t = (a ,a ) ; j ?^ k, t = 0; i 7^ X, t — 1 , Theorem ^,6, A necessary ojid. sufficient condition that the local n-cloment problem be solvable when n > 2 rnd N(a ) holds for all i, is that i | (a ,a ) I | 9^0. Proof. By the previous theorem I |(a .a ) I I =0 im^Dlies a latticetheoretic relation among the ja 1, Othen'^ise,

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-55|xl + Z s^(a^,x). For sufficiently close to \e.^\ , u is a norm aiid u(a^) = laj ; i 1, and u(a^) = r^. Repeating this norm trpjisformation for each X gives the desired result. Corollary ^.2^. The local n-olement problem is solvable for a subset of neutral elements vrlUch are equilateral under some norm, P£oof. If II [a^] holds and r is the common dist^jice, then I |(a^,a^)| ^= (-r)^^-l(n 1) ^ 0 for n > 2. Im3£, i.lZ. 2 r |a I 9^ 0 implies 2 r = 0. i i 1 ^ i Proof. u(x) = |:c| + i being a norm implies Z r = Definition 4.2. 2 rjaj 0 is said to be reducible if there exist {s^} such that S s | a | = o < Z | s | or E | r i = 0, Definition 4.1. S rjaj ^ o\s said to be irreducible If it is not reducible. Corollarv If ,2^. 2 r J aj 0 < Z I r J is Irreducible if a^d only if no principal minor, of order exceeding tvro, of I I (a ,a ) i I vrjilshes. CoroUar;^ 4.26. If 2 rjaj ^0 is Irreducible, then ?^ 0 for all i and a^ = a^ if a^id only if i = j. £2i3ma ^.11. I a J 0 < Z I r J may be expressed as a set of irreducible equations. Proof. If Z rjaj ^ 0 < Z [ r^ j is reducible, it may be expressed in terms of txTO equations of lesser degree. iiffia 4.12. If n(a^) holds for all i and Z r i a | Si o is irreducible, then there is s r,uch that sr^ is L Integer

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-56for all 1. Proof , ir(a^) holding implies N(a'') vxhioh implies the existence of a norm, | 1, such that ] a'' | is a:i integer. Then (a ,a ) are Integers and are integers nroportional to the r , k Lemma ^.20, If J r^ja^l ^0 is irreducible and the are Integers v:ith greatest common denominator . 1 , then the I r^i^ I are imique • Proof . Non-uniqueness of the r^ gives rise to a reduction of the equation. n m Definition li.±, E r J a, | ^ I s, |b, | is said to be in 1=1 1 1=1 ^ 1 canonical form if it is irreducible and the r^^ and s^ are positive Integers vrlth greatest common denominator,!. Lemma '1. 21, Z s. | a^ | ^ 0 end s > 0, for all 1. imrily ill i"" > " Sj^ = 0, for all 1. Lemma ^.22. S s^|a^i ^ rjb] ; r,s^ > 0, for all 1, ajid N(b) holding imply a^ = b, for all 1. Proo f. E s^(a^,b) = r(b,b) = 0. Lemma i.2i. E s^|a^| -J^lb^l + r \^^\ , s^,r^,r^ > 0, and II(b ,b ) holding imply B(b ,a ,b ) , 2 1 i 2 Remark ^,1, ExnmDles in v;liich E r. |a | 0 is 1 ^ i irreducible mr.y be constructed for n > 2, and also vfith Il(a ) holding^ for n > 'l, n ra Remark 4,2. For S s la I E r [b I in canonical form 1=1 i 1 1=1 i i and m,n > 2, no betv/eenness relations among the a .b need i' 1 exist as shovjn by the follov/ing example;

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-57In the set algebra on 9 elements "^2»°2*°3 '^1**^2*^3* ^1*^2*^3' ^2 = i\*y\*^y^^U = a, = {o ,0 ,d ,d e }, b = {c ,c ,d ,d ,e }. 3 12123 3 23133 Let Ic^l = |d^l = |e^l = 1/2; i = 1,2,3. Then (a^,aj) = 3 = (b ,b.); 1 7^ J, and (a ,b ) = 2. Prom these distances one obtains |a^| + \a^\ + |a^| |b^| + |b^l + [b^l irreducibly, but no betweenness relations (although "seml-betv:eenness" conditions subsist; e.fT,, a Aa^ 0 such that the follov/ing four conditions are pairv;ise equivalent, la| + |b| Si |o| + |d| Q(a,b;c,d) q|a| + r|b| ^ sjcj + t|d| aO^ = ©Od Proof . q|a| + r|b| ^ sjc| + t|d| implies B(a,c,b) , B(a,d,b), B(o,a,d), B(c,b,d) implies Q(a,b;c,d) vrhich implies aO b = c0d which in turn implies |ai + |b| ^ |aAb| + |a V b| |c A d| + |c V d| fii |c| + |d| . Corollary ^.27. II[a,b,c,d] holding implies that I I (x,y)l I = 0 if and only if a,b,c,d form a pseudolinear quadru^ole, (This is a viealcer form of Corollary 2,4-.) Corollary ^.2g. H(a,b,c,d,e,f ) holding implies that Q(a,d;b,c) and Q(o,f;d,e) together imply Q(a,e;b,f).

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"5^ Proof , |al ibl ^ \o\ |d| S [e] If], (This Is an alternate proof of Theorem 2,6.) Lerama ^.2^, If N(a,b,e,d,e) subsist?, then 2|a| + |b| Si !c| + ld| + je! is equivalent to B(a,c,b), B(a,d,b), B(a,e,b), R(c,a,d), B(c,a,o), B(d,e,o), (b,o) = (d,e), (b,d) = (c,e), rjid (b,e) = (o,d). Proof . By solution of the five disfcrjice equations: 2(a,x) + (b,x) (c,x) + (d,x) + (e,x); x = a,b,c,d,e. Remark 'j-.3. An exf'.mple v/ith 2|a| + |bl ^ |c| + |d| + |e| may be realized in the set algebra on 3 elements f,r:,h where a = {f,g}, b = {h}, c = {f}, d = {e}, and e = {f,g,h}. Lemma ^1.35. |a| + jdj a |bl + \ o\ and |b| + |dl a I el + |f| if ond. only if ia| + 2|dl |o| + |e| + |f| and 2lb| + |cl ^ |al + |e| + |f|. Corollary . 29 « In a metric lattice, Q(a,d;b,c) rjnd Q(b,d;e,f) imply B(a,e,d), B(a,f,d), B(b,e,c), B(b,f,c), B(c,d,f), B(e,d,c), B(a,b,f), B(n,b,e), (a,c) = (e,f), (a,e) = (c,f), sua (a,f) = (c,e)j vriien II(a,b,c,d ,e,f ) subsists.

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CHAPTER V ADDITIONAL PROBLEIIS One of the outstrjidlnG imsolved problems in metrio lattice theory is a suitable characterization of lattices which are metric. Several algebraic conditions aro necesGary: (1.) The Ir.ttice must be modular, (2.) Every cloain must form a metric lattice; that is, be order isomorphically imbeddable in the real number system (separability), (3.) in every bounded distributive sublattice each set of elements {a^} such that i ^ J implies a^ J a^ = a (fixed a) must be countable, (^.) In any bounded sublattice all asoendins or dencendins sequences of projectively related projective roots (29) must be finite. The set of all finite subsets of an uncountable set uith the addition of an upper bo.aid satisfies (1), (2), a^id (^0, but not (3). A famous unsolved problem is Souslin's problem. In one form it conjectures tliat any chain containing no uncountable collections of non-overlappinc intervals forms a metric lattice (1, 16). If, given a distributive lattice, we imbed it in a relatively complemented lattice (l4) ?jid (3) holds for the resultPJit lattice, then every set of disjoint chain intervals is countable and the validity of Souslin's conjecture -59-

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-6oIraplies (2). Hence (2) and (3) are not necessarily independent in relatively complemented distributive lattices. All finite (but not all countable) nodular lattices . may be norraed. Also the Boolepji alGObra [2^°] (set of all subsets of a coim table set) is metric; which includes the norraine of all coim tabic distributive lattices and the lattice structure of real separable Ililbert space. The bounded free distributive lattice amd multiple projective root lattice generated by an arbitrary set are nornable. Necessary and sufficient conditions for the norraing of bounded distributive lattices are laioxm indirectly through extension to
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-61Btrong Isotonlclty for come of the elements j cej^eralize the lattice or the norm to iDosets; use a parameter (time) dependent norm and, hence, a varying metric. One simple generalization f olloi;s : A /O-comiected set is a set v;ith a symmetric relation such tlir.t for every pair a,b in the set there exists a finite chain of elements from a to b connected by the relation (3), After asslf^ning positive real numbers to each unordered pair a,b vfnore a ^ b, \tq nay define the dlstajice {a,b) as the greatest lovrer bound of the sum of the dlstrjices across the ^-connections from a to b; resulting in a quasi-metric space. Given a poset with a real valued function u on It, vie may define a^ b if nnd only if a is comparable ulth b rn.d let |(|u(a)| iu(b)|)| be the distp>ice associated with a^ b. A metric lattice is an example of such a quasi-metric poset liavlng a, a ^ b,b as a minimal path. n Kf?2iy of the theorems and lomraas are valid vrhen Z oo Is ret)laced by Z et cetera. Some, hoviever, must be 1=1 modified. For instsjice. In transforming 2 sJa I into Z B''la''l 1 1' i' J ' ' the strong convergence criterion 11m s 2^ = 0 is sufficient, 1-»
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-62Dedelclnd cuts in the a^-chain since it mo,y not be tjell ordered, q may even be 0 in the case of lattices v.'ithout lovrer bounds. Infinite determinants, of course, present their ovm problems. There may be some application in analysis for these trrnsformations in the infinite case. The set algebra on a coimtable set normed by i{a^}| = 2"^ provides some Interesting expjnples as it induces a liighly discontinuous function on the real line, and the ro^l separable Hilbert space. The a"' of Convention ^-,2 constitute a maximal chain in the free distributive lattice cenerated by the a^. Is there any convenient useful method for indexing chp,ins (or the vrhole lattice) in free lattices (5)? Somevrhnt related is the problem of determining distributive structure in lattices generated by {b.^} such that D(a«). This problem Pnd its generalization hinge on the case of four generators. Host of the imsolved problems concerning pseudo-linear quadruples are concerned with some pha.se of the complex structure of the free modular lattice r^onerated by a,b,o,d vrhere aO b = ' cO cl (2). Quasi-distributivity vrould have more sir:?ilf icnnoe if it oould be shovm equivalent to some lattice theoretic identity (2^1-) (finite if such exists; othertfise infinite), or its forbidden pictures could be suitably clir.racterized. Finite exajnples of forbidden pictures with '4nondistributlvities and 2 or 3 pairiiigs are readily found. Examples irith 2 or 3 non-distributivities or only one pairing may

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.63not bo found go readily if suoh erdlEst, The follovring may be of some help In approaching such problems, A metric measure of a property is a non-negative real valued function uhoce vajiishinr; is equivalent to the validity of the property in question, (a,b) is a metric measure of the property a = b, (a,a J b) of the property a ^ b, and (a,b,c) of the property D(a,b,c). Letting a» = a ^ (c ^ d) and b» = V A b J (c )[ d) V7e have (a !) b,c J d) (a« J b»,c J d) = (a J b, a 5 b J (c ;( d)) > 0 and (a b,c I a) (a« J b',c )[ d) = (a A 1^, (a ;( b) J (c ;( d)) (l/2)(a,b,c ^ d) = (a )( o d,b ^ c]^d) (ajb,ajbj (cjjd)). In addition, lecting a^ = a 0 (c d), b^ = b 5 (c )( d), c^= o, .d^ = d, a^^^ = a^ J A ^i^»^^fl ; \ V A \)> Vi = °i V ^\ A \+l " ^i V A ^ ^ i^I^lies a < a^ < a^ < (a J c) J( (a w d), and symnetrically. Also a^ f) b = c, d, is equiv' 1 V j_ i V i ^ alent to a^ = a^^^,b^ = b^^^,c^ = o^_^^, a_nd d^ = d^^^ P2id this is equivalent to a^ ~ ^± ~ ^ j ' °i ~ = ^ » J > i. Note that the a »s et oete-a are defined recursively 1 by multiple use of the (») trr-ns format ion. Hence a ,b .c .d A A i' i' i' i metrically approach the condition ajb^c^data rate V V depending on their uetrio approximation to the other properties shorn. If L(a,b,c,d) has no infinite chains (nuch as 1^1 a finite or distributive lattice (5)) or if the lattice is comiolete (1), (metric completeness is necescary for conver.-.enoe

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in measure) the sequence generates r."'" ,0''' such that a"*" = o"'^ V a^ jb"^ ,c^ jd^are not necessarily distinct (e.g. the prime projective root lattice: a ^ b = b ^ c and b = d) , By the use of these transformations aO b = c Od may be approjcimr.ted and forbidden pictures may be Generated, Since distinctness of elements is not precorvod, this method in its present form does not give a recursive (possibly infinite) identity equivf.lent to quasi-distributivity. There are several otudies of pseudo-linear quadruples both in and out of metric lattices vrhich might be made paralleling those for metric betvreonness. One is the charr-.cterIzation of pseudo-linear quadruples similar to that (of Vald) for metric betTjoenness (3). Another is the systematic study of their transitivities rjid other clonoly related proi^erties (21, 25, 36, 32). One might define B»(a,b,c) if and only if B(a,b,c) rnd (a,b) < (b,o), njid Q'(a,b;c,d) if and only if Q(a,b;c,d) ojid (a,b) < (c,d) (ordered betueenness oiid pseudolinear quadruples) and determine wliich of the trct:>isitivlties are valid, . There are several unsolved ouestions in connection x;ith the n-eloment problem, the simplest of v:liich is determining v;hether or not the term "local" may be omitted from Theorem h,6. The general n-element problem oven for the case n = 2 is tmsolved (latticetheoretically) but as previously remarked, any necessary cjid sufficient conditions may become

PAGE 68

-65quite conplicr.ted, 'riiere is also, of course, the aiinlocous problem for a countably (or arbitrarily) infinite set; adding converccnce criteria. It alco lir.s bearing on the problem of cliaracterlziri^ metric Ir^ttioes, A nimple criterion for isotonlcity of the troiisformations of the type u(x) = |x| + E s^(a^,::) is not Imoim nnd may be closely related to the n-clement problem. The composition of the c^oup G (Convention 3,";^) or its immediate ceneralizr.tions (o.c. its extension to a
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-6bTransformation on the sublattlce cenerr.ted hy {a^} may be realized as a sequence of trans formrt ions of the type u(x) = |x| + r(a,x); hence the conclusion. Do the coefficients snd combinatorial properties of Z s. ja. I =0 vrhen expressed in oaiionioal form, have ajialocues j_ i 1 in number theory otsI classical algebra? Is there any feasible method for determining, in f^eneral, the possible coefficients and the related lattice-theoretic conditions associated vrith these constraints. One might devise a general method, based on the modulrr lav;, of syntlieslzing these constraints or tr-nsforming them to lower order. Given a set {a^} of neutral elements, can one determine the least upper bound of the coefficients ajid the sum of their absolute values for all possible constraints in csjionical form vrith the sajae value of n? Tliis is a set-theoretic problem as the constraints correspond to pairs of collections of subsets of the {a^}, each collection having each a^ the same number of times. These questions, of course, all have analogies in the geometry of metric lattices.

PAGE 70

,C p •H -67^ > > > O O O O^-' > > > > > O ^ (1) ^ 'd'— 'd 'd — ' 'd^< —"d-d >>> — ^-d -d <<< ooO'-^<<:< coo ^^-^ o o o o <<<<<<< ^ ^ ^ ^ ^ ^ ^ > -d -d < 'd >^ 0) 'd^< ^-d 'd < < < o > > ^ o c O O C O— >>>>>>> ^ ^ ^ j=i 05 03 cfl cd cC cs 'cJ OS K) d cj 03 c C oj c o; a cd ci c; "§ d d « > > >>>>>>>>>>>>>> 0) < -d -d --^ > > > o o o o < < < < > ® -d < -d < < < o o o < < < ^ < < < Cu d C3 > C9 0> < OS CM II as < 05 VI H H as O O II a) II CVJ €5 U «> > at > > > 03 > t:) 'd > > > > •d > > d o §^ > d > d— ' d > d d o > d < ^ > d o > > © d > > d < > -d -d ^ > > d > > d d <«—>_^ '-^<: < > o o ,o > > — ' c d > ^^^ 05 "d fd "d > > > d d d — < <^ ^ jQ > > > d d d-^ © > d o > d > > ^ o o > > d o d d— '> ^v^< d < < rQ ,Q > — > > d ^ d d^> d © ^ > > d '->• < > -d -d d > > --^ d d <-^^ '-^< < o— > o o d > :> d d < — ^ d > > d d © < d © < d o < d < d 'd < 'd t:) < < d d > >. o o < < d d > > < < d d o < d < d < o d < ^ d > • — — > d < ^ d © < d X2 < d rd-— ^ < 'd t:) d < < d d ^> > < .o d < < d d < -< o o d < < d rj > < ^ ;3 d < < ^ d d p © < < d © < < d < < d -d -d < < < ;d ^ ^ < < < d d d-^ > > ^ o o o < < < .o o ^ ,D < < < < d p d d^< © < o < < d © < -d 'd < < c o < < < < d d •H Lr\=^ inr^LfNdu^,CMLf\zh irNr^ir\=h LnH L^vdlJr^^^l^ ^^^^^^^^^^^^^^^

PAGE 71

GLOSSAIIY OP SPECIAL SYI-IBOLS^ Letter convention ... C,1.3 jj |j^ { j ^-1.5 » Index convention ...C-l.-l I I ... C-1 D-1 ^ V» <• A» ^ » <,>,<»>... C-1.7, G-l.g, abc, abc, Hbc ... C-I.13 C-1.10, D-i l e, $ ... c-1.19. D-1.15 A, V ... [2^] ... C-1.21 ... c-1.9, a^, u(x), vU) ... C-1.23 9^ ... C-J^.ll u, V, w ... t ... C-2.3 ^ a. a ... C-3.3,'C-ii-.^, dim ( ) ... C-1.25 Z' ... cJi.S f, C, li ... (,)... C-1.11, D-1.7 R,b^c,a ... C-2.Ii, D-1.2 (,,)... C-1.16 ^i' ^-^+-2 ( ; . ) ... c-3.1 ^-'-^ AC ] ... C-I.2 ^0 ^--'-^ B( , , ) ... c-1.12 js, u ... ) c-i.iii, c-1.15 r ... c-'i.io H( ) ... c-1.17, c-i.is ^rs ^-•^•5 lU ; ) ... C-3.2 ^ijk^ C-'^'l^ L( ) ... C-.1.22 Q( , ; , ) ... c-2.1 1 C nuiabers refer to conventions, D niiabers refer to definitions. -6g-

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BIBLIOGRAPHY 1. Garrett Blrkhoff , Lattice Theory (Revised Edition ) . Am, Kath. Soo,, New York, ISW, 2. Garrett Blrl. 11, N. Jacobsen, Lectures In Abstract Alg:ebra . Vol. I, Vejci Nostrand, New York, 1951, " 12, L. M. Kelly, The /Geometry of normal lattices . Duke Math. Jour., vol, 19"7l952), pp, ^1 670.

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-7013. S, Kiss, Trangformatlons on Lg.ttloes and Stniot^u*es of Lopclc . Privately Printed, Nev; York, 19^7. 1^. H. H, KaoNellle, Extension of a distributive lattice to a Boolean rlnp: . Bull. Am, Math. Soc,, vol. 45 (1939) i PP. '^52 ^55. 15, D. Kaharam, Aa alp:ebralo characterization of measure al^^ebrns . Annals of Math., vol. (1947) , pp. l^H167 16, D, Maharam, Set functions and Sous 11ns hypothes^ . B ,. Dull. Am, Math. Soc,, vol. 5^ (19^2!), pp. 5S7 590. 17, K. Monger, Untersuchen uber allpemelne metrlk . Math, Annalen, vol. 100 (192J?) , pp. 75 I63. 1^, J. von Neumann, Lectures on Continuous Qeometrles . Vol., I III, Princeton Univ. Press, Princeton, 1937* 19. 0, Ore, On the foundations of abstract algebro, Annals of Math., vol. 36TT935) , PP. ^6 WT* 20. B. J, Pettis, Notes on Meas^are Theory , Part Lattices and Lattice Functlonals'^IIimeo^^rRphed ) . Tulane University, New Orleajis, 1951. 21. E. Pitcher and M, P. Smiley, Transltiylties of betweenness. Trans, Am. Math, Soo,, vol, 52 (19'^-2) , PP. 95 m. 22. B, C, Rennie, The T heory of Lattices . Poister cmd Jaggj Cambridge, England, 1951V 23. M. P. Schutzenberger, Construction du trellis engendr^ pf}.r duex elements et une cliain finie discrete, Corapte Rendus (Paris) , vol, 235T1952) , pp. 926 92g, 2^, M, P. Schutzenberger, Sur certain axiomes de la th6orie des structures . Corapte Rendus ( Paris ) , vol , 221 TT9'^5), PF. 21^ 222. 25, M. P. Smiley and W, R, Trejisue, Applications of transitivities of betweenness in lattice tueory. Dull, Am, Math, Soc, voi7rr9 (l^il3), pp.Tgo 2^7,

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-7126. K, P. Smiley, A comparison of alfjebralc, me trip . and lattice betweennees . Bull. /on. Math. Soc., vol. ^9 ( 19^3 ) i pp. 246 252. 27. M. F. Smiley, extension of metric distributive lattices with an application to f-^enera l analysis . Trans, Am. Hath. Soc, vol. 5fc (19^4), Pp7^35 ^77. 2g, R, M. Thrall ejid D. 0. Duncan, Note on free modular lattices . Am. Jour, of Math., vol. 75 ( 19^3 ) » PP. 627 632. 29, R, M. Thrall, On the pro.leotlve structure of a modular lattice . Proc. Am. Math, Soc, vol. 2 (1951) • ~ pp. 146 152. 30. P. M. VJhitman, Pree lattices . Annals of Math., vol. 42 (19^1), pp. 1^6 152. 31. P. M. Whitman, Free lattices II . Annals of Math., vol. 43 (19^2), pp. 109 115. 32, L, R. Wilcox BXid. M. P. Smiley, Metric lattices . Annals of Math., vol. 4o (1939), PP. 309 327.

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DIOGRAPHICAL SICETCH Alfred Baker Lehman was bom March 21, 1931 in Clevelajid, Ohio, He \JV8 graduated from Cranbrook School, Dloomfield Hills, Michigan in June 19^7. His undergraduate work was taken at Case Institute of Technology, Western Reserve University, rnd Ohio University from which he received the Bachelor of Science degree in February, 1950, His graduate work V7as taken at Ohio State University, the University of Chicago,, and the University of Florida, -72-

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This dissertation v;as prepared imder the direction of the chairman of the candidate's supervisory committee and has been approved by all members of the committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council and v/as approved as partial fulfilment of the requirements for the decree of Doctor of Philosophy, January 30, 19 5^+ Dean, College of Arts and Sciences Dean, Graduate School SUPERVISORY COMMI'i?TEE: Chairman