Residuation Theory PDF
Preview Residuation Theory
RESIDUATION THEORY T. S. Blyth University of St. Andrews M. F. Janowitz University of Massachusetts PERGAMON PRESS OXFORD . NEW YORK . TORONTO SYDNEY . BRAUNSCHWEIG Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1972 T. S. Blyth; M. F. Janowitz All Rights Reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Ltd. First edition 1972 Library of Congress Catalog Card No. 77-142177 Printed in Germany 08 016408 0 PREFACE THE aim of the present volume is to add a substantial contribution to the textbook literature in the field of ordered algebraic structures. The fundamental notion which permeates the entire work is that of a residu- ated mapping and this is indeed the first unified account of this topic. The origin of this concept has been traced by J. Schmidt [26] to M. Benado [3] and G.Nöbeling [22, 23]. It also appears in the work of P.Dubreil [11] and R. Croisot [8]. The general theory of residuated mappings seems to have lain dormant for approximately 20 years until the appearance of papers by J. C. Derderian [9] and M. F. Janowitz [16,17] ; however, during this time particular types of residuated mappings were employed in studying residuated semigroups, principally by M. L. Dubreil-Jacotin [12], LMolinaro [20], J.Querré [24] and T.S.Blyth [5]. This text has grown out of courses given by T.S.B. at the Universities of St. Andrews, Western Australia and Western Ontario and by M.FJ. at the Universities of Massachusetts, New Mexico and Western Michigan. In this (hopefully happy) marriage of our efforts, the choice of text mate- rial has, quite frankly, been selfish and more or less motivated by our own research interests. It was never our intention to write an encyclopaedia on the subject (we leave that happy task to someone else!) but rather to produce a self-contained and unified introduction to the subject which may be used either as a textbook or as a reference book in this area. In this connection we mention that many research papers are listed in the bibliography without explicit reference to their contents being made in the text. Many of these have had to be excluded because of space limitations and we hope that we have offended no one by so doing. The reader will undoubtedly find the present text useful in supplying the unified back- ground material necessary to read those papers. Little attempt has been vu vin PREFACE made to credit results to their originators and we have tried to present the material in a well-marshalled and readable manner without the clutter of numerous references. The advantage of the combined efforts of a British and an American author is that the book is designed to satisfy a variety of courses on both sides of the Atlantic. For example, Chapter 1 may be used as an advanced under-graduate course on ordered sets and lattice theory; Chapters 1 and 2 as a one-semester post-graduate course on lattice theory; and the whole text as an M.Sc. course on lattices and residuated semigroups. We have included a large number of illustrative examples and exercises. The exer- cises are of varying degrees of difficulty, some serving to provide examples and counter-examples to supplement the text, some being designed to help the student gain intuition and some to extend the text material. We assume that the reader has most of his under-graduate training behind him, so that he has a good grounding in abstract algebra; for example, we shall feel free to assume that the reader knows what is meant by a ring, an ideal of a ring, etc. We shall also assume that he is familiar with Zorn's axiom. Some knowledge of general topology will be helpful for some of the examples, but not essential for understanding the book. Though no prior knowledge of lattice theory is expected, the reader might find it helpful on occasion to consult a standard elementary text on the subject, for there will be times when we simply will not be able to delve as deeply as we would like into a given branch of the subject. We have organized the text by dividing it into three chapters, the first of which contains an introduction to residuated mappings and lattice theory. This chapter has been specifically written with an advanced undergraduate course in mind and contains all of the elementary material which is required later. In Chapter 2 we deal with the concept of a Baer semigroup and employ residuated mappings to show how these semi- groups may be used to study lattices. In so doing, we incorporate some of the important work of D. J.Foulis [13] and S.S.Holland, Jr. [15] on orthomodular lattices. Finally, in Chapter 3, we use the notion of a residuated mapping as a basis for a discussion of residuated semigroups. In particular, we show how a certain residuated semigroup plays a funda- mental rôle in the study of homomorphic images of ordered semigroups, PREFACE ix a starting point of which is a result of A. Bigard [4]. Whenever possible, we have phrased our results in terms of residuated mappings; for this reason, even one well versed in lattice theory would find here a fresh approach to the subject. By far the majority of the results given here appear for the first time in book form and indeed some of them are only just seeing the lightf of day. Most of the material in Chapters 2 and 3 has been developed in the last decade; we hope that it may serve to inspire further research. Our grateful thanks are due to Professors E.A.Schreiner and G.D. Crown for their valuable criticisms of the manuscript and to Dr T. P. Speed for his assistance in the proof-reading. Finally we would express our admiration at the ease with which the printers undertook a difficult task. T.S.B.;M.F.J. CHAPTER 1 FOUNDATIONS 1. Ordered sets Let E be a set and let R be a binary relation between elements of E. Of the properties which R may enjoy, the most commonly encountered in mathematics are the following: R is said to be (a) reflexive if (Vx e E) xRx; (b) transitive if (xRy and }>lte) => xRz; (c) antisymmetric if (xRy and ;yito;) => x = y; (d) symmetric if xity => jita. A relation i£ which satisfies (a), (b) and (d) on 2? is called an equivalence relation on E, as the reader will undoubtedly be aware. Although we shall meet with many equivalence relations in the pages which follow, we shall be concerned primarily with relations which satisfy the properties (a), (b), (c). A relation which satisfies these three properties on E will be called an order relation on E or simply an ordering on E. By an ordered set we shall mean a set E together with an ordering on it. We shall usually denote an ordering by the symbol < so that the properties (a), (b), (c) become (a) (\/xeE)x < x; (b) (x < y and y < z) => x < z; (c) (x < y and y < x) => x = y. Upon occasions, however, we shall find it convenient to use a variation of this symbol. EXAMPLE 1.1. The only binary relation on a set E which is both an equivalence relation and an ordering on E is the relation of equality. EXAMPLE 1.2. The set R of real numbers is an ordered set, < having 1 2 RESIDUATION THEORY its usual meaning. For each subset A of R we shall use the notation A+ = {x e A; x > 0}. EXAMPLE 1.3. The set P(E) of all subsets of a set E is an ordered set under the relation £ of set inclusion. EXAMPLE 1.4. The set Z+ of positive integers is an ordered set under the definition m =^ n o m is a factor of n. EXAMPLE 1.5. Let E, F be sets with F ordered. The set Map (E,F) of all mappings of E into F is ordered under the definition f<goQ/xeE) f(x)<g(x). EXAMPLE 1.6. If El9...9En are ordered sets, then so also is their n Cartesian product X E under the definition t (x ,...,x )<(j ,...,j )o(/= l,...,w) Xi <yt. 1 n 1 n More generally, if (E ) is a family of ordered sets then X E is an (X (XeA a <xeA ordered set with respect to the ordering specified in Example 1.5: We shall say that a set E is totally ordered (or forms a cAtfm) if it is ordered in such a way that for any given elements x, y e E we have x < y or y < x. The ordered set of Example 1.2 is totally ordered. If we define , , \ Λ x < yo(x < y and x Φ y), then a totally ordered set may be described as an ordered set in which any two distinct elements x, y satisfy either x < y or y < x. It should be noted, however, that the relation < as defined above is not an ordering for it does not satisfy (a). Also, the relations x $ y and y < x are equivalent only in the case of a totally ordered set and not in the general case of an ordered set. We say that two elements x, y of an ordered set E are comparable if x < y or y < x and denote this symbolically by writing xj/fy. If, on the other hand, neither x < y nor y < x holds, then we say that x, y are incomparable and write x \\ y. If-Eis an ordered set and F is a non-empty subset of E, then we shall say that F is totally unordered if ths elements of FOUNDATIONS 3 F are pairwise incomparable. This is equivalent to saying that in F we have x < yox = y;in other words, the restriction to F of < is equality. EXAMPLE 1.7. Consider the set E = {a, b, c}. In the ordered set P(E) the subset J — {{a}, {b} {c}} is totally unordered as is the subset 9 K={{a,b},{a,c},{b,c}}. Let R be a binary relation on a set E and let R* denote its converse, i.e. R* is given by xR*y o yRx. It is readily seen that if R is an ordering on E, then so also is R*. We denote the converse of < by > and the converse of < by >. Many ordered sets can be conveniently represented by means of Hasse diagrams. In such a diagram we represent x < y by /■ ox i.e. we join the point representing x to that representing y by an increasing line segment. In drawing Hasse diagrams we shall agree not to include any superfluous line segments which arise through the transitivity of <. This principle is illustrated in the following example. EXAMPLE 1.8. Let E be the set of all positive factors of 12. If we order E according to Example 1.2 we obtain a chain. If we order E ac- cording to Example 1.4, the corresponding Hasse diagram is 12 o 1 Note that in the above diagram we have not joined 2 and 12 by a direct line segment; for 2 < 6 and 6 < 12 imply that 2 < 12 by transitivity, so such a line is superfluous. 4 RESIDUATION THEORY By the dual of a Hasse diagram we mean the Hasse diagram associated with the converse ordering. It is clear that to obtain the dual diagram all we have to do is to turn the original upside down. By the dual of an ordered set we shall mean the same set equipped with the converse order. When we the need arises, we shall use the notation P* to denote the dual of the ordered set P. EXERCISES 1.1. Prove that every finite ordered set has a Hasse diagram. 1.2. Let E be the set of factors of 120, ordered by divisibility. Draw the Hasse dia- gram for E. 1.3. Draw the Hasse diagrams for all possible orderings on a set consisting of (a) 3, (b) 4, (c) 5 elements. 1.4. Let (Ρχ, <i), (P , < ) be ordered sets. Show that the relation < defined on 2 2 PxPby 1 2 (xi, x) < 0>i, y2) o (*i ^ 1 J>i and x < 2 ^2) 2 2 is an ordering. Show also that < is a total ordering if and only if < 1 and < are total 2 orderings and Pi or P consists of a single element. 2 1.5. Let (Ρχ, <i), (P , < ) be ordered sets. Prove that the relation < defined on 2 2 PixP by 2 (xi,x)<(yuy)~[Qithct * < ιΛ ^ 2 2 A lor xi = yi and * ^2^2 2 is an ordering. This is known as the lexicographic ordering on P x P . Show that < is x 2 a total ordering if and only if <i and < are total orderings. 2 1.6. Let P and P be the ordered sets with respective Hasse diagrams x 2 O 1 ,/\ 0 o o Draw the Hasse diagrams for P xP and ΡχΡχ when ordered as in Examples 1.4 x 2 2 and 1.5. 1.7. Let (Pi, <i) and (P , < ) be disjoint ordered sets. Show that each of the 2 2 following defines an ordering on P = Pi u P : 2 (a) x < yox <ij> or x< y; 2 (b) x < y o x <! y, x < y or χεΡχ and yeP. 2 2 FOUNDATIONS 5 2. Mappings between ordered sets; residuated mappings Let P be an ordered set. By an {order) ideal of P we shall mean any non- empty subset / of P satisfying the property (xel and y<x)=>yel. By a principal ideal of P we shall mean any ideal of the form [*->*] = {yeP;y < x}. We define the dual notions of an (order) filter of P to be any non-empty subset J having the property (x 6 J and y > x)=> y e «/, and a principal filter of P to be any filter of the form [>,-►] = {JG?;J> *}. If P, Q are ordered sets and/: P -* ß is any mapping, then for each non-empty subset R of Q we define the pre-image ofR under f to be the subset of P given by f*-(R) = {xeP;f(x)eR}. Our first result shows how the above notions can be used to character- ize an important type of mapping between ordered sets. THEOREM 2.1. If A, B are ordered sets andf: A-> B is any mapping, the following conditions are equivalent: (l)x<y~f(x)<f(y); (2) the pre-image of every principal ideal of B is either empty or is an ideal of A; (3) the pre-image of every principal filter of B is either empty or is a filter of A. Proof Note that we use the same symbol < to denote the ordering in both A and B; no confusion will arise since the context will always make it clear to which set we are referring. We shall show that (1) o (2) ; a dual argument will clearly yield (1) <=> (3). Suppose then that/satisfies (1) and let x e B be such that/" [<-, x] φ 0 (where we write/*" [«-, x] in place of/*" ([<-, x])). Then if y ef*~ [<-, x] and z<jwe have, by (1), f(z)
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