Applied abstract algebra

worked-The text should be useful to mature mathematics students, to students in computer or information science with an interest and background knowledge in algebra, and to physical scie

Rudolf Lidl Gunter Pilz

Applied Abstract Algebra

Second Edition

With 112 illustrations

Springer

F.W Gehring Mathematics Department East Hall

University of Michigan Ann Arbor, MI 48109 USA

K.A Ribet Department of Mathematics University of California

at Berkeley Berkeley, CA 94720-3840 USA

Mathematics Subject Classification (1991): 05-01, 06-01, 08-01, 12-01, 13-01, 16-01, 20-01, 68-01, 93-01

Library of Congress Cataloging-in-Publication Data

Lidl, Rudolf

Applied abstract algebra I Rudolf Lidl, Gunter Pilz - 2nd ed

p em - (Undergraduate texts in mathematics)

Includes bibliographical references and index

ISBN 0-387-98290-6 (he : alk paper)

1 Algebra, Abstract I Pilz, Gunter, 1945- II Title

III Series

512'.02-dc21

© 1998 Springer-Verlag New York, Inc

All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden

The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone

ISBN 0-387-98290-6 Springer-Verlag New York Berlin Heidelberg SPIN 10632883

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Armstrong: Basic Topology

Armstrong: Groups and Symmetry

Axler: Linear Algebra Done Right

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Banchoff/Wermer: Linear Algebra Through

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Berberian: A First Course in Real Analysis

Bremaud: An Introduction to Probabilistic

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Browder: Mathematical Analysis: An

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Childs: A Concrete Introduction to Higher

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Chung: Elementary Probability Theory with

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Curtis: Linear Algebra: An Introductory

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Dixmier: General Topology

Driver: Why Math?

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Exner: An Accompaniment to Higher

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Fischer: Intermediate Real Analysis

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Fleming: Functions of Several Variables Second

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Hilton/Holton/Pedersen: Mathematical Reflections: In a Room with Many Mirrors

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Isaac: The Pleasures of Probability

Klambauer: Aspects of Calculus

Lang: A First Course in Calculus Fifth edition Lang: Calculus of Several Variables Third edition Lang: Introduction to Linear Algebra Second edition

Lang: Linear Algebra Third edition

Lang: Undergraduate Algebra Second edition

Lang: Undergraduate Analysis

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(continued after index)

Preface

Algebra is beautiful It is so beautiful that many people forget that algebra

can be very useful as well It is still the case that students who have studied

mathematics quite often enter graduate studies or enter employment

without much knowledge of the applicability of the algebraic structures

they have studied

The aim of this book is to convey to senior undergraduate students,

graduate students, and lecturers/instructors the fact that concepts of

ab-stract algebra encountered previously in a first algebra course can be used

in many areas of applications Of course, in order to apply algebra, we

first need some theory which then can be applied Hence we tried to

blend the theory and its applications so that the reader can experience

both parts

This book assumes knowledge of the material covered in a course on

linear algebra and, preferably, a first course in (abstract) algebra covering

the basics of groups, rings, and fields, although this book will provide the

necessary definitions and brief summaries of the main results that will

be required from such a course in algebra

This second edition includes major changes to the first edition,

pub-lished in 1984: it contains corrections and, as we believe, substantial

improvements to the first four chapters of the first edition It includes a

largely new chapter on Cryptology (Chapter 5) and an enlarged chapter

on Applications of Groups (Chapter 6) An extensive Chapter 7 has been

added to survey other (mostly quite recent) applications, many of which

Vll

Vlll P_r_e_fa_c_e _

were not included in the first edition An interdependence chart of the material in the sections is presented below

For a one-semester course (2-3 hours per week) on Applied Algebra

or Discrete Mathematics, we recommend the following path: §§1, 2, 3, 4, 6-17, 21, 22, 23, and selected topics in Chapter 7 chosen by the instructor

As in the first edition, we again emphasize the inclusion of out examples and computational aspects in presenting the material More than 500 exercises accompany the 40 sections A separate solution man-ual for all these exercises is available from the publisher The book also includes some historical notes and extensive references for further reading

worked-The text should be useful to mature mathematics students, to students

in computer or information science with an interest and background knowledge in algebra, and to physical science or engineering students with a good knowledge in linear and some abstract algebra Many of the topics covered are relevant to and have connections with computer science, computer algebra, physical sciences, and technology

It is a great pleasure to acknowledge the assistance of colleagues and friends at various stages of preparing this second edition Most of all, we would like to express our sincere appreciation to Franz Binder, who prepared many drafts and the final version of the entire book with U\TEX Through his expertise in algebra, he was able to suggest many improvements and provided valuable information on many topics Many useful suggestions and comments were provided by: E Aichinger (Linz, Austria), G Birkenmeier (Lafayette, Louisiana), J Ecker (Linz, Aus-tria), H E Heatherly (Lafayette, Louisiana), H Kautschitsch (Klagenfurt, Austria), C J Maxson (College Station, Thxas), W B Muller (Klagen-furt, Austria), G L Mullen (University Park, Pennsylvania), C Nobauer,

P Paule (Linz, Austria), A P J Vander Walt (Stellenbosch, South Africa), and F Winkler (Linz, Austria) Special thanks are due to L Shevrin and I 0 Koryakov (Ekaterinenburg, Russia) for preparing a Russian translation of the first edition of our text Their comments improved the text substantially We also wish to thank Springer-Verlag, especially

Mr Thomas von Foerster, Mr Steven Pisano, and Mr Brian Howe, for their kind and patient cooperation

Childs (1995), Herstein (1975), Jacobson (1985), and Lang (1984) Application-oriented

books include Biggs (1985), Birkhoff & Bartee (1970), Bobrow & Arbib (1974), Cohen,

Giusti & Mora (1996), Dorninger & Muller (1984), Fisher (1977), Gilbert (1976), Prather

(1976), Preparata & Yeh (1973), Spindler (1994), and Stone (1973) A survey of the

present "state of the art" in algebra is Hazewinkel (1996) (with several more volumes

to follow) Historic notes on algebra can be found in Birkhoff (1976) Applications of

linear algebra (which are not covered in this book) can be found in Goult (1978), Noble

& Daniel (1977), Rorres & Anton (1984), and Usmani (1987) Lipschutz (1976) contains

a large collection of Exercises Good books on computational aspects ("Computer

Al-gebra") include Geddes, Czapor & Labahn (1993), Knuth (1981), Lipson (1981), Sims

(1984), Sims (1994), and Winkler (1996)

IX

7 Switching Circuits 55

Notes 93

XI

14 Irreducible Polynomials over Finite Fields

15 Factorization of Polynomials over Finite Fields Notes

5 Cryptology

21 Classical Cryptosystems

22 Public Key Cryptosystems

23 Discrete Logarithms and Other Ciphers Notes

7 Further Applications of Algebra

28 Semigroups

29 Semigroups and Automata

30 Semigroups and Formal Languages

31 Semigroups and Biology

32 Semigroups and Sociology

33 Linear Recurring Sequences

34 Fast Fourier Transforms

XVI List of Symbols

Lattices

CHAPTER

In 1854, George Boole (1815-1864) introduced an important class of

al-gebraic structures in connection with his research in mathematical logic

His goal was to find a mathematical model for human reasoning In

his honor these structures have been called Boolean algebras They are

special types of lattices It was E Schroder, who about 1890 considered

the lattice concept in today's sense At approximately the same time,

R Dedekind developed a similar concept in his work on groups and

ideals Dedekind defined, in modern terminology, modular and

distribu-tive lattices, which are types oflattices of importance in applications The

rapid development of lattice theory proper started around 1930, when

G Birkhoff made major contributions to the theory

Boolean lattices or Boolean algebras may be described as the richest

and, at the same time, the most important lattices for applications Since

they are defined as distributive and complemented lattices, it is natural to

consider some properties of distributive and complemented lattices first

§1 Properties and Examples of Lattices

We know from elementary arithmetic that for any two natural

num-bers a, b there is a largest number d which divides both a and b, namely

the greatest common divisor gcd(a, b) of a and b Also, there is a smallest

1

number m which is a multiple ofboth a and b, namely the least common multiple m = lcm(a, b) This is pictured in Figure 1.1

Thrning to another situation, given two statements a, b, there is a

"weakest" statement implying both a and b, namely the statement "a

and b," which we write as a 1\ b Similarly, there is a "strongest" statement which is implied by a and b, namely "a or b," written as a v b This is pictured in Figure 1.2

A third situation arises when we study sets A, B Again, there is a largest set contained in A and B, the intersection An B, and a smallest one containing both A and B, the union AU B We get the similar diagram

in Figure 1.3

It is typical for modern mathematics that seemingly different areas lead to very similar situations The idea then is to extract the common features in these examples, to study these features, and to apply the resulting theory to many different areas This is very economical: proving one single general theorem automatically yields theorems in all areas to which the theory applies And usually we discover many more new areas

of application as the theory builds up

We are going to do exactly that for the three examples we have met above In the first two chapters, we shall study collections of items which have something like a "greatest lower bound" and a "least upper bound";

we shall call them lattices Before doing so, we need a short "warm-up" to get fit for the theory to come

One of the important concepts in all of mathematics is that of a tion Of particular interest are equivalence relations, functions, and order relations Here we concentrate on the latter concept and recall from an introductory mathematics course:

rela-§1 Properties and Examples of Lattices 3

-~~ ~ -~ -Let A and B be sets A relation Rfrom A to B is a subset of A x B, the

cartesian product of A and B Relations from A to A are called relations

on A, for short If (a, b) E R, we write aRb and say that "a is in relation

R to b." Otherwise, we write aJlb If we consider a set A together with a

relationR, we write (A,R)

A relation R on a set A may have some of the following properties:

R is reflexive if a R a for all a E A;

R is symmetric if aRb implies bRa for all a, bE A;

R is antisymmetric if aRb and bRa imply a= b for all a, bE A;

R is transitive if aRb and b R c imply aRc for all a, b, c E A

A reflexive, symmetric, and transitive relation R is called an

equiva-lence relation In this case, for any a E A, [a] := {b E A I aRb} is called the

1.1 Definition A reflexive, antisymmetric, and transitive relation R

on a set A is called a partial order (relation) In this case, (A, R) is called a

partially ordered set or poset

Partial order relations describe "hierarchical" situations; usually we

write :::: or <; instead of R Partially ordered finite sets (A, :S) can be

graphically represented by Hasse diagrams Here the elements of A are

represented as points in the plane and if a :::: b, a#- b (in which case we

write a < b), we draw b higher up than a and connect a and b with a line

segment For example, the Hasse diagram of the poset (P({l, 2, 3}), <;)is

shown in Figure 1.4, where P(S) denotes the power set of S, i.e., the set of

1.2 Definition A partial order relation :::: on A is called a linear order

if for each a, b E A either a :::: b or b :::: a In this case, (A, :S) is called a

For example, ({1, 2, 3, 4, 5}, :S) is a chain, while (P({1, 2, 3}), s;) is not

If R is a relation from A to B, then R-1 , defined by (a, b) E R-1 iff (b, a) E R, is a relation from B to A, called the inverse relation of R If (A, :S) is a partially ordered set, then (A,::::_) is a partially ordered set, and ::::_ is the inverse relation to ::::

Let (A, :S) be a poset We say, a is a greatest element if "all other elements are smaller." More precisely, a E A is called a greatest element of

A if for all x E A we have x :::: a The element b in A is called a smallest element of A if b :::: x for all x E A The element c E A is called a maximal element of A if "no element is bigger," i.e., c :::: x implies c = x for all

x E A; similarly, d E A is called a minimal element of A if x :::: d implies

x = d for all x E A It can be shown that (A,::::) has at most one greatest and one smallest element However, there may be none, one, or several maximal or minimal elements Every greatest element is maximal and every smallest element is minimal For instance, in the poset of Figure

is no greatest element

§1 Properties and Examples of Lattices 5

-~ ~ -~ -1.3 Definitions Let (A, :S) be a poset and B <;A

(i) a E A is called an upper bound of B if b :::: a for all b E B

(ii) a E A is called a lower bound of B if a :::: b for all b E B

(iii) The greatest amongst the lower bounds of B, whenever it exists, is

called the infimum of B, and is denoted by inf B

(iv) The least upper bound of B, whenever it exists, is called the

For instance, if (A, :S) = (IR., :S) and B is the interval [0, 3), then inf B =

0 and sup B = 3 Thus the infimum (supremum) of B may or may not be

an element of B If B' = N, inf B' = 1, but supB' does not exist

If B = {a1, , an}, then we write inf(a1, , an) and sup(a1, , an)

instead of inf{a1, , an} and sup{a1, , an}, respectively

The following statement can neither be proved nor can it be refuted

(it is undecidable) It is an additional axiom, that may be used in

math-ematical arguments (we usually do so without any comment), and it is

equivalent to the Axiom of Choice

1.4 Axiom (Zorn's Lemma) If (A, ::::) is a poset such that every chain of

elements in A has an upper bound in A, then A has at least one maximal

element

In general, not every subset of a poset (L, :S) has a supremum or

an infimum We study those posets more closely which are axiomatically

required to have a supremum and infimum for certain families of subsets

1.5 Definition A poset (L, :S) is called lattice ordered if for every pair

x, y of elements of L their supremum and infimum exist

1.6 Remarks

(i) Every chain is lattice ordered

(ii) In a lattice ordered set (L, :S) the following statements can be easily

seen to be equivalent for all x and y in L:

(a) x:::: y;

(b) sup(x, y) = y;

There is another (yet equivalent) approach, which does not use order

relations, but algebraic operations instead

1 7 Definition An (algebraic) lattice (L, /\, v) is a set L with two

bi-nary operations 1\ (meet) and v (join) (also called intersection or product

Sometimes we read x v y and x /\ y as "x vee y" and "x wedge y."

The connection between lattice ordered sets and algebraic lattices is as follows

1.8 Theorem

(i) Let (L, :S) be a lattice ordered set If we define

x /\ y := inf(x, y), x v y := sup(x, y),

then (L, A, v) is an algebraic lattice

(ii) Let (L, A, v) be an algebraic lattice If we define

X :S y :{=::} X /\ y = X, then (L, :S) is a lattice ordered set

Proof

(i) Let (L, :S) be a lattice ordered set For all x, y, z E L we have:

(Ll) x /\ y = inf(x, y) = inf(y, x) = y /\ x,

x v y = sup(x, y) = sup(y, x) = y v x

(L2) x /\ (y /\ z) = x /\ inf(y, z) = inf(x, inf(y, z)) = inf(x, y, z)

= inf(inf(x, y), z) = inf(x, y) /\ z = (x /\ y) /\ z,

and similarly x v (y v z) = (x v y) v z

(L3) x /\ (x v y) = x /\ sup(x, y) = inf(x, sup(x, y)) = x,

x v (x /\ y) = x v inf(x, y) = sup(x, inf(x, y)) = x

(ii) Let (L, A, v) be an algebraic lattice Clearly, for all x, y, z in L:

• x /\ x = x and x v x = x by (L4); sox:::: x, i.e., ::::is reflexive

§1 Properties and Examples of Lattices 7

-~ ~ -~ -• If x :::: y andy :::: x, then x 1\ y = x andy 1\ x = y, and by (L1)

x 1\ y = y 1\ x, sox= y, i.e., ::::is antisymmetric

• If x:::: y andy:::: z, then x 1\ y = x andy 1\ z = y Therefore

X = X 1\ y = X 1\ (y 1\ z) = (X 1\ y) 1\ Z = X 1\ z,

sox:::: zby (L2), i.e.,:::: is transitive

Letx,y E L Thenx/\(xvy) = ximpliesx:::: xvyandsimilarlyy:::: xvy

If z E L with x :::: z andy :::: z, then (x v y) v z = x v (y v z) = x v z = z

and so x v y :::: z Thus sup(x, y) = x v y Similarly inf(x, y) = x 1\ y

1.9 Remark It follows from Remark 1.6 that Theorem 1.8 yields a

one-to-one relationship between lattice ordered sets and algebraic lattices

Therefore we shall use the term lattice for both concepts The number ILl

of elements of L denotes the cardinality (or the order) of the lattice L

If N is a subset of a poset, then V xEN x and /\xEN x denote the

supre-mum and infisupre-mum of N, respectively, whenever they exist We say that

the supremum of N is the join of all elements of N and the infimum is

the meet of all elements of N

In Definition 1.7, we have seen that for each of the laws (L1)-(L3),

two equations are given This leads to

1.10 (Duality Principle) Any "formula" involving the operations 1\ and v

which is valid in any lattice (L, /\, v) remains valid if we replace 1\ by v

and v by 1\ everywhere in the formula This process of replacing is called

dualizing

The validity of this assertion follows from the fact that any formula

in a lattice that can be derived using (L1 )-(L3) remains correct if we

interchange 1\ and v, :::: and :::::, respectively, everywhere in the formula,

because every dual of a condition in (L1 )-(L3) holds, too This is very

convenient, since we only have to prove "one-half' of the results (see,

e.g., 1.13 and 1.14)

1.11 Definition If a lattice L contains a smallest (greatest) element

with respect to ::::, then this uniquely determined element is called the

0 and 1 are called universal bounds If they exist, Lis called bounded

Every finite lattice L is bounded (see Exercise 6) If a lattice is bounded

(by 0 and 1), then every x in L satisfies 0:::: x:::: 1, 01\ x = 0, 0 v x = x,

1 1\ x = x, 1 v x = 1 We consider some examples of lattices

8 1 Lattices

-~~ -Set :::: xl\y xvy 0 1

Theorem 1.8 and Remark 1.9 enable us to represent any lattice as

a special poset or as an algebraic structure using operation tables In Figures 1.8 and 1.9, we present the Hasse diagrams of all lattices with at most six elements Vf denotes the ith lattice with n elements Figure 1.6 shows an example of a poset which is not a lattice (since sup(b, c) did not exist)

In Figure 1.10, we give the operation tables for the lattice V~, which table all x 1\ y and x v y, for x, y in the lattice Observe that all entries in these tables must again belong to the lattice

1.13 Lemma In every lattice L the operations 1\ and v are isotone, i.e.,

y :S Z ==} X 1\ y :S X 1\ Z and X V y :S X V Z

Proof y:::: z ==} xA.y = (x A X) 1\ (y A.z) = (xA.y) 1\ (x A.z) ==} x A.y:::: x/\z

1.14 Theorem The elements of an arbitrary lattice satisfY the following

distributive inequalities:

x 1\ (y v z) ::::_ (x 1\ y) v (x 1\ z),

x v (y 1\ z) :::: (x v y) 1\ (x v z) (1.1)

§1 Properties and Examples of Lattices g

-~~ ~ -~ -Proof From x /\ y :::: x and x /\ y :::: y :::: y v z we get x /\ y :::: x /\ (y v z),

and similarly x /\ z :::: x /\ (y v z) Thus x /\ (y v z) is an upper bound for

both x /\ y and x /\ z; therefore x /\ (y v z) ::::_ (x /\ y) v (x /\ z) The second

We can construct new lattices from given ones by forming

sub-structures, homomorphic images, and products

1.15 Definition A subsetS of a lattice Lis called a sublattice of L if S

is a lattice with respect to the restrictions of/\ and v from L to S

Obviously, a subset S of L is a sublattice of the lattice L if and only

if Sis "closed" with respect to /\ and v (i.e., s1, Sz E S ==} s1 /\ Sz E S

and s1 v s 2 E S) We note that a subset S of a lattice L can be a lattice

with respect to the partial order of L without being a sublattice of L (see

Example 1.16(iii) below)

§1 Properties and Examples of Lattices

(i) Every singleton of a lattice L is a sublattice of L

(ii) For any two elements x, y in a lattice L, the interval

[x,y] :={a ELI x _:::a_::: y}

is a sublattice of L

(iii) Let L be the lattice of all subsets of a vector space V and letS be the

set of all subspaces of V Then Sis a lattice with respect to inclusion

but not a sublattice of L

1.17 Definitions Let L and M be lattices A mapping f: L -+ M is

called a:

(i) join-homomorphism iff(x v y) = f(x) v f(y);

(ii) meet-homomorphism if f(x 1\ y) = f(x) 1\ f(y);

(iii) order-homomorphism ifx _::: y ==} f(x) _::: f(y);

hold for all x, y E L We call fa homomorphism (or lattice homomorphism)

if it is both a join- and a meet-homomorphism Injective, surjective, or

bijective (lattice) homomorphisms are called (lattice) monomorphisms,

L toM, thenf(L) is called a homomorphic image of L; it is a sublattice of M

(see Exercise 11) If there is an isomorphism from L to M, then we say

that Land Mare isomorphic and denote this by L ~ M

It can be easily shown that every join-(or meet-)homomorphism is

an order-homomorphism The converse, however, is not true (why?)

The relationship between the different homomorphisms is symbolized in

Figure 1.11

11

FIGURE 1.11

u

For example, and s V t are isomorphic under the

morphism 0 f-+ r, a f-+ s, b f-+ t, 1 f-+ u The map 0 f-+ r, a f-+ t, b f-+ s,

1 f-+ u is another isomorphism Observe that in Figures 1.8 and 1 9 we have in fact listed only all nonisomorphic lattices of orders up to 6 Ob-serve that there are already infinitely many different lattices with one element As another example, vf is isomorphic (but not equal) to 1 ~ We see, in most cases it makes sense to identify isomorphic lattices

1.18 Example Let L1 , L2 , and L 3 be the lattices with Hasse diagrams

of Figure 1.12, respectively We define:

§1 Properties and Examples of Lattices 13

However, f is not a homomorphism, since

Dually, g is a join-homomorphism, but not a homomorphism his neither

a meet- nor a join-homomorphism, since

h(a1 1\ b1) = h(01) = 03 and h(a1) 1\ h(b1) = a3 1\ b3 = a3,

h(al v b1) = h(11) = 03 and h(a1) v h(b1) = a3 v b3 = b3

1.19 Definition Let L and M be lattices The set of ordered pairs

{ (x, y) I x E L, y E M}

with operations v and 1\ defined by

(xl,Yl) V (xz,Yz) := (x1 V Xz,Yl V Yz), (xl,Yl) 1\ (xz,Yz) := (xl /\Xz,Yl 1\Yz),

is the direct product of L and M, in symbols L x M, also called the product

two lattices

It is verified easily that L x M is a lattice in the sense of Definition 1 7

The partial order of L x M which results from the correspondence in

1.8(1.8) satisfies

(1.2)

1.20 Example The direct product of the lattices L and M can

graph-ically be described in terms of the Hasse diagrams at the top of the

following page

2 Give an example of a poset which has exactly one maximal element but does not have a greatest element

3 Let (IR, ::S) be the poset of all real numbers and let A = {x E IR I x 3 < 3} Is there an upper bound (or lower bound) or a supremum (or infimum) of

Exercises 15

Define a partial order <; on C as in Equation 1.2 by: x 1 + iy 1 <; x 2 + iy 2 if

and only if x1 ::::; x 2 and y1 ::::; y 2 Is this a linear order? Is there a minimal or

a maximal element in (C, <;)? How does <; compare with the lexicographic

order ::::; defined by x 1 + iy 1 ::::; x 2 + iy 2 if and only if x 1 < x 2 , or x 1 = x 2 and

y, ::S Yz?

9 An isomorphism of posets is a bijective order-homomorphism, whose

in-verse is also an order-homomorphism Prove: Iff is an isomorphism of a

poset L onto a poset M, and if L is a lattice, then M is also a lattice, and f

is an isomorphism of the lattices

a closed interval [a, b] and let D[a, b] be the set of all differentiable functions

on [a, b] Show by example that D[a, b] is not a sublattice of C[a, b]

L is isomorphic to a sublattice of M

Figure 1.9)?

(D(k), gcd, lcm) is a lattice Construct the Hasse diagrams of the lattices

D(20) and D(21), find isomorphic copies in Figures 1.8 and 1.9, and show

16 Let C1 and C2 be the finite chains {0, 1, 2) and {0, 1}, respectively Draw

16 1 Lattices

-~~~ -17 Let L be a sublattice of M and let f: M + N be a homomorphism If M is

§2 Distributive Lattices

We now turn to special types of lattices, with the aim of defining very

"rich" types of algebraic structures, Boolean algebras

2.1 Definition A lattice Lis called distributive if the laws

x v (y !\ z) = (x v y) !\ (x v z),

x !\ (y v z) = (x !\ y) v (x !\ z),

hold for all x, y, z E L These equalities are called distributive laws

Due to Exercise 9, the two distributive laws are equivalent, so it would

be enough to require just one of them

2.2 Examples

(i) (P(M), n, U) is a distributive lattice

(ii) (N, gcd, lcm) is a distributive lattice (see Exercise 4)

(iii) The "diamond lattice" V~ and the "pentagon lattice" Vl are not tributive: In V~, a v (b !\c)= a#- 1 = (a v b)!\ (a v c), while in vl,

These are the smallest nondistributive lattices

2.3 Theorem A lattice is distributive if and only if it does not contain a sublattice isomorphic to the diamond or the pentagon

A lattice which "contains" the diamond or the pentagon must clearly

be nondistributive The converse needs much more work (see, e.g., Szasz (1963)) As an application of 2.3 we get:

§2 Distributive Lattices 17

-~ -2 4 Corollary Every chain is distributive lattice

2.5 Example The lattice with Hasse diagram

sub lattice

2.6 Theorem A lattice Lis distributive if and only if the cancellation rule

X 1\ y = X 1\ z, X V y = XV Z ==} y = Z holds for all X, y, Z E L

Proof Exercise 6

2 7 Definition A lattice L with 0 and 1 is called complemented if for

each x E L there is at least one element y such that x 1\ y = 0 and x v y = 1

Each such y is called a complement of x

(iii) Not every lattice with 0 and 1 is complemented For instance, a in

~ ~ does not have a complement In fact, every chain with more

than two elements is not complemented

(iv) The complement need not be unique: a in the diamond has the two

complements b and c

(v) Let L be the lattice of subspaces of the vector space IR.2 If T is a

complement of a subspace S, then S n T = { 0} and S + T = IR.2 Hence

a complement is a complementary subspace If dimS = 1, then S

has infinitely many complements, namely all subspaces T such that

S EB T = IR.2 Therefore L cannot be distributive, as the following

theorem shows

18 1 Lattices

-~~~ -2 9 Theorem and Definition IfL is a distributive lattice, then each x E L has at most one complement We denote it by x'

Proof Suppose x E L has two complements y1 and y 2 Then x v y1 = 1 =

x v Yz and x 1\ Y1 = 0 = x 1\ Yz; thus Y1 = Yz because of 2.6 D Complemented distributive lattices will be studied extensively in the following sections

2.10 Definition Let L be a lattice with zero a E Lis called an atom if

a # 0 and if for all b E L : 0 < b :::=: a ==} b = a

2 11 Definition a E L is called join-irreducible if for all b, c E L

a = b v c ==} a = b or a = c

Otherwise a is calledjoin-reducible

2.12 Lemma Every atom of a lattice with zero is join-irreducible

Proof Let a be an atom and let a= b v c, a# b Then a= sup(b, c); so

2.14 Definitions If x E [a, b] = {v E L I a :::=: v :::=: b} andy E L with

x 1\ y = a and x v y = b, then y is called a relative complement of x with respect to [a, b] If all intervals [a, b] in a lattice L are complemented, then

L is called relatively complemented If L has a zero element and all [0, b]

are complemented, then L is called sectionally complemented

Exercises

1 Prove the generalized distributive inequality for lattices:

a', then

a v (a' 1\ b) = a v b

§3 Boolean Algebras 1 9

-~ -~~ -3 Which of the lattices in Figures 1 8 and 1.9 are distributive?

Comple-mented?

4 Show that the set N, ordered by divisibility, is a distributive lattice Is it

functions from S to D is a distributive lattice, where f ::S g means f(x) ::S g(x)

for all x

distributive lattices are again distributive

(Cf Exercises 1.5 and 1.17.)

elements are join-irreducible?

sectionally complemented), show that this applies to Mas well

§3 Boolean Algebras

Boolean algebras are special lattices which are useful in the study of

logic, both digital computer logic and that of human thinking, and of

who showed that fundamental properties of electrical circuits of bistable

elements can be represented by using Boolean algebras We shall consider

such applications in Chapter 2

3.1 Definition A complemented distributive lattice is called a Boolean

Distributivity in a Boolean algebra guarantees the uniqueness of

complements (see 2.9)

20 1 Lattices

-~~~ -3.2 Notation From now on, in Chapters 1 and 2, B will denote a set with the two binary operations 1\ and v, with zero element 0 and a unit element 1, and the unary operation of complementation', in short

B = (B, 1\, v, 0, 1,') orB= (B, 1\, v), or simply B

3.3 Examples

(i) (P(M), n, U, 0, M,') is the Boolean algebra of the power set of a set

M Here n and U are the set-theoretic operations intersection and union, and the complement is the set-theoretic complement, namely

M \A = A'; 0 and M are the "universal bounds." If M has n( E N0) elements, then P(M) consists of 2n elements

(ii) Let IR be the lattice Vf, where the operations are defined by

Then (IR, /\, v, 0, 1,') is a Boolean algebra If n E N, we can turn IBn

into a Boolean algebra via 1.19:

(il, · · · 1 in) 1\ (h, · · · ,jn) := (il 1\h, · · · 1 in 1\jn),

(il, 1 in) V (h, ,jn) := (il V jl, 1 in V jn),

D

§ 3 Boolean Algebras 21

-~ -~~ -3.5 Corollary In a Boolean algebra B we have for all x, y E B,

x ::::; y {::=::} x' ::::: y'

Proof x::::; y {::=::} x v y = y {::=::} x' 1\ y' = (x v y)' = y' {::=::} x' ::::-_ y' D

3.6 Theorem In a Boolean algebra B we have for all x,y E B,

X :S y {::=::} X 1\ y' = 0 {::=::} X 1 V y = 1 {::=::} X 1\ y = X

{::=::} XV y = y

Proof See Exercise 4

3 7 Definition Let B1 and B 2 be Boolean algebras Then the mapping

f: B1 -+ B 2 is called a (Boolean) homomorphism from B1 into B 2 iff is a

(lattice) homomorphism and for all x E B we have f(x') = (f(x) )'

Analogously, we can define Boolean monomorphisms and

isomor-phisms as in 1.17 If there is a Boolean isomorphism between B1 and B 2 ,

we write B1 ~b B 2 The simple proofs of the following properties are left

to the reader

3.8 Theorem Let f: B1 -+ B2 be a Boolean homomorphism Then·

(i) f(O) = 0, f(1) = 1;

(ii) for all x, y E B1, x ::::; y ==} f(x) :S f(y);

(iii) f(B1 ) is a Boolean algebra and a "Boolean subalgebra" (which is defined

as expected) of B2

3.9 Examples

(i) If M c N, then the map f: P(M) -+ P(N); A r-+ A is a lattice

monomorphism but not a Boolean homomorphism, since for A E

P(M) the complements in M and N are different Also, f(1) = f(M) =

M #-N = the unit element in P(N)

(ii) If M = {1, , n}, then {0, l}n and P(M) are Boolean algebras, and

the map f : {0, l}n -+ P(M), (i1, , in) r-+ {k I ik = 1} is a Boolean

isomorphism It is instructive to do the proof as an exercise

(iii) More generally, let X be any set, A a subset of X, and let

22 1 Lattices

-~~~ -3.10 Theorem Let L be a lattice Then the following implications hold:

(i) Lis a Boolean algebra==} Lis relatively complemented;

(ii) Lis relatively complemented==} Lis sectionally complemented;

(iii) L is finite and sectionally complemented ==} every nonzero element a

of L is a join of finitely many atoms

Proof

(i) Let L be a Boolean algebra and let a:::: x:::: b Define y := b /\(a v x')

Then y is a complement of x in [a, b], since

x /\ y = x /\ (b /\ (a v x')) = x /\ (a v x') = (x /\ a) v (x /\ x') = x /\ a = a

and

x v y = x v (b A (a v x')) = x v ((bAa) v (b Ax')) = x v (b Ax')

= (x v b) A (x v x') = b A 1 = b

Thus L is relatively complemented

(ii) If Lis relatively complemented, then every [a, b] is complemented; thus every interval [0, b] is complemented, i.e., L is sectionally complemented

(iii) Let {p1, ,Pn} be the set of atoms:::: a ELand let b = P1 v · · · v Pn· Now b :::: a, and if we suppose that b #- a, then b has a nonzero

complement, say c, in [0, a] Letpbe an atom:::: c, thenp E {pl, ,Pn}

and thus p = p /\ b :::: c /\ b = 0, which is a contradiction Hence

Finite Boolean algebras can be characterized as follows:

3.11 Theorem (Representation Theorem) Let B be a finite Boolean bra, and let A denote the set of all atoms in B Then B is isomorphic to P(A),

alge-i.e.,

(B, A, V) ~b (P(A), n, U)

Proof Let v E B be an arbitrary element and let A(v) :={a E A I a :::: v}

Then A(v) <;A Define

h: B -+ P(A); v f-+ A(v)

We show that h is a Boolean isomorphism First we show that h is a

Boolean homomorphism: For an atom a and for v, w E V we have

here, the second equivalence follows from 3.6 Note that h(O) = 0 and

0 is the unique element which is mapped to 0 Since B is finite we are

able to use Theorem 3.10 to verifY that his bijective We know that every

v E B can be expressed as ajoin of finitely many atoms: v = a1 v ···van

with all atoms ai :::; v Let h(v) = h(w), i.e., A(v) = A(w) Then ai E A(v)

and ai E A(w) Therefore ai.:::; w, and thus v.:::; w Reversing the roles of v

and w yields v = w, and this shows that h is injective

1b show that h is surjective we verifY that for each C E P(A) there is

some v E B such that h(v) = C Let C = {c1, , Cn} and v = c1 v · · · v Cn

Then A(v) 2 C, hence h(v) 2 C Conversely, if a E h(v), then a is an atom

with a.:::; v = c1 v · · · v Cn Therefore a.:::; ci, for some i E {1, , n}, by 2.12

and 2.13 So a= ci E C Altogether this implies h(v) = A(v) =C D

3.12 Theorem The cardinality of a finite Boolean algebra B is always of

the form 2n, and B then has precisely n atoms Any two Boolean algebras with

the same finite cardinality are isomorphic

Proof The first assertion follows immediately from 3.11 If B1 and B 2 have

the same cardinality m E N, then m is of the form 2n, and B1, B 2 have

both n atoms So B1 ~b P({1, , n}) ~b B 2 by 3.9(ii), hence B1 ~b B 2 D

In this way we have also seen:

3.13 Theorem For every finite Boolean algebra B #- {0} there is some

n EN with

24 1 Lattices

-~~~ -3.14 Examples

(i) The lattice of the divisors of 30, i.e., the Boolean algebra B =

to 30), has 8 = 23 elements and is therefore isomorphic to the lattice

of the power set P({a, b, c})

(ii) We sketch the Hasse diagrams of all nonisomorphic Boolean algebras

of orders < 16:

0

I <>

[BO =Vi [Bl = vf [f£2 = v{ [f£3

3.15 Remark The identification of an arbitrary Boolean algebra with

a power set as in 3.11 is not always possible in the infinite case (see Exercise 9) Similar to the proof of 3.11 it can be shown that for every (not necessarily finite) Boolean algebra B there is a set Manda Boolean monomorphism ("Boolean embedding") from B to P(M) This is called

Stone's Representation Theorem

3.16 Definition and Theorem Let B be a Boolean algebra and let X

be any set For mappings f and g from X into B we define