This appendix provides a very concise summary of the results from topology needed in Chapter 11 and its exercises. Our account aims solely to pinpoint those topological ideas we need. Any standard text may be consulted for proofs and further motivation. Our references are to W.A. Sutherland, An Introduction to Metric and Topological Spaces.

Topology is usually introduced as an abstraction of concepts first met in elementary analysis, such as open neighbourhood and continuous function. In a topological space, a family of open sets generalizes the open neighbourhoods of the euclidean spaces ℝn. The axioms for a topological space bring under topology's umbrella many structures which are very unlike euclidean spaces. It is certain of such spaces that concern us. The metric spaces which utilize the idea of a distance function analogous to the modulus function on ℝ and which are frequently used as a stepping stone to topological spaces play no role here.

Proving results in topology demands a certain facility in manipulating sets and maps. The formulae set out in, pp. xi–xiii, are a necessary stock in trade.

Topological spaces. A topological space (X; T) consists of a set X and a family T of subsets of X such that

1. (T1) ø ∈ T and X ∈ T,

2. (T2) a finite intersection of members of T is in T,

3. (T3) an arbitrary union of members of T is in T.

There are many examples in mathematics of statements which, overtly or covertly, assert the existence of an element maximal in some ordered set (commonly, a family of sets under inclusion). The first section of this chapter addresses the question of the existence of maximal elements. This question cannot be answered without a discussion of Zorn's Lemma and the Axiom of Choice, and this necessitates an excursion into the foundations of set theory. It would be inappropriate to include here a full discussion of the role and status in mathematics of Zorn's Lemma and its equivalents. Rather we seek to complement the treatment in set theory texts of this important topic and, although our account is self-contained, it is principally directed at readers who have previously encountered the Axiom of Choice. En route, we provide belated justification for the arguments in 2.39, prove some intrinsically interesting results about ordered sets, and derive the results on prime and maximal ideals on which the representation theory in Chapter 11 rests. Those who do not wish to explore this foundational material but who do wish to study Chapter 11, may without detriment, skip over the first section of this chapter; see 10.15.

Do maximal elements exist? – Zorn's Lemma and the Axiom of Choice

Aside from the treatment of ordinals, ordered sets have traditionally played a peripheral role in introductory set theory courses.

In Chapter 2 we began an exploration of the algebraic theory of lattices, armed with enough axioms on ∨ and ∧ to ensure that each lattice 〈L; ∨, ∧〉 arose from a lattice 〈L;≤〉 and vice versa. Now we introduce identities linking join and meet which are not implied by the laws (L1)–(L4) and their duals (L1)∂–(L4)∂ defining lattices (recall 2.9). These hold in many of our examples of lattices, in particular in powersets. In the second part of the chapter we abstract a different feature of powersets, namely the existence of complements.

Lattices satisfying additional identities

Before formally introducing modular and distributive lattices we prove three lemmas which will put the definitions in 4.4 into perspective. The import of these lemmas is discussed in 4.5.

Lemma.Let L be a lattice and let a,b,c ∈ L. Then

1. (i) a ∧ (b ∨ c) ≥ (a ∧ b) ∨ (a ∧ c), and dually,

2. (ii) a ≥ c implies a ∧ (b ∨ c) ≥ (a ∧ b) ∨ c, and dually,

3. (iii) (a ∧ b) ∨ (c) ∨ c) ∧ (a) ≤ (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a).

Proof. We leave (i) and (iii) as exercises. (Alternatively, see Exercise 2.9.) By the Connecting Lemma, (ii) is a special case of (i).

This new edition of Introduction to Lattices and Order is substantially different from the original one published in 1990. We believe that the revision greatly enhances the book's usefulness and topicality. Our overall aims however remain the same: to provide a textbook introduction which shows the importance of the concept of order in algebra, logic, computer science and other fields and which makes the basic theory accessible to undergraduate and beginning graduate students in mathematics and to professionals in adjacent areas.

In preparing the new edition we have drawn extensively on our teaching experience over the past 10 years and on helpful comments from colleagues. We have taken account of important developments in areas of application, in particular in computer science. Almost all the original material is included, but it has been completely re-organized. Some new material has been added, most notably on Galois connections and fixpoint calculus, and there are many new exercises.

Our objectives in re-arranging the material have been:

• to present elementary and motivational topics as early as possible, for pedagogical reasons;

• to arrange the chapters so that the first part of the book contains core material, suitable for a short, first course;

• to make it easy for particular interest groups to pick out just the sections they want.

Order, order, order – it permeates mathematics, and everyday life, to such an extent that we take it for granted. It appears in many guises: first, second, third, …; bigger versus smaller; better versus worse. Notions of progression, precedence and preference may all be brought under its umbrella. Our first task is to crystallize these imprecise ideas and to formalize the relationship of ‘less-than-or-equal-to’. Besides presenting examples and basic properties of ordered sets, this chapter also introduces the diagrams which make order theory such a pictorial subject and give it much of its character.

Ordered sets

What exactly do we mean by order? More mathematically, what do we mean by an ordered set?

Order. Each of the following miscellany of statements has something to do with order.

1. (a) 0 < 1 and 1 < 1023.

2. (b) Two first cousins have a common grandfather.

3. (c) 22/7 is a worse approximation to π than 3.141592654.

4. (d) The planets in order of increasing distance from the sun are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto.

5. (e) Neither of the sets {1, 2, 4} and {2,3,5} is a subset of the other, but {1,2,3,4,5} contains both.

6. (f) Given any two distinct real numbers a and b, either a is greater than b or b is greater than a.

Lattices of congruences play a central role in lattice theory and in algebra more widely. This chapter develops the rudiments of a theory which goes way beyond the scope of an introductory text such as this.

Introducing congruences

In group theory courses it is customary, after homomorphisms have been introduced, to go on to define normal subgroups and quotient groups (factor groups) and to reveal the intimate connection between these concepts that is summed up in the fundamental Homomorphism Theorem (also called the First Isomorphism Theorem). We begin with a summary of the basic group theory results, expressed in a form that will make the parallels with the lattice case stand out clearly. This summary is prefaced by a brief refresher on equivalence relations.

Equivalence relations: a recap. We recall that an equivalence relation on a set A is a binary relation on A which is reflexive, symmetric and transitive. We write a ≡ b (mod θ) or a θ b to indicate that a and b are related under the relation θ we use instead the notation (a, b) ∈ θ where it is appropriate to be formally correct and to regard θ as a subset of A × A.

An equivalence relation θ on A gives rise to a partition of A into non-empty disjoint subsets.

In previous chapters we have introduced various classes of lattices. We have given examples of members of these classes, and described some of their general properties. We now turn our attention to structure theorems. Later (see Chapters 10 and 11) we give a concrete representation, as a lattice of sets, of any (bounded) distributive lattice. This chapter deals, less ambitiously, with the finite case, and reveals a very satisfactory correspondence between finite distributive lattices and finite ordered sets. We show that any finite distributive lattice L can be realized as a lattice O(P) of down-sets built from a suitable subset P of L. We begin by discussing in general terms the problem of finding a subset of a lattice L which, as an ordered set, uniquely determines L.

Building blocks for lattices

This chapter draws on the final section of Chapter 2, concerned with join-irreducible elements. In the remarks below we also make links with the material on concept lattices from Chapter 3, but familiarity with this is not essential for the representation theory that follows.

Remarks on lattice-building. In 2.42 we defined a non-zero element x of a lattice L to be join-irreducible if x = a ∨ b implies x = a or x = b for all a, b ∈ L. Proposition 2.45 showed that if L satisfies (DCC), and hence certainly if L is finite, the set J(L) of join-irreducible elements of L is join-dense; that is, that every element of L can be obtained as a (possibly empty) join of elements from J(L).