The Role of Order in Lattice Theory

Abstract

As an algebraic system, a lattice is a two operation system with equation definable axioms. The usual axiom set is the following:

• Idempotent laws: a ∨ a = a, a ∧ a = a

• Commutative laws: a ∨ b = b ∨ a, a ∧ b = b ∧ a

• Associative laws: (a ∨ b) ∨ c = a ∨ (b ∨ c), (a ∧ b) ∧ c = a ∧ (b ∧ c)

• Adsorptive laws: a ∨ (a ∧ b) = a, a ∧ (a ∨ b) = a.

A particularly significant feature of a lattice as an algebraic system is the fact that it has an intrinsic order relation given by $a \leqslant b{\text{ if and only if }}a \wedge b = a$ or equivalently $a \leqslant b{\text{ if and only if }}a \wedge b = b$

The lattice axioms insure that this relation is a partial order consistent with the lattice operations.

The intrinsic character of the lattice ordering sets lattices apart from most other ordered algebraic systems, such as ordered groups, ordered rings, ordered fields, etc. in which the order relation is imposed on the system and has specified properties. Furthermore, the lattice operations themselves can be defined in terms of the lattice order relation.

$$\begin{gathered} {\text{Least upper bound: }}a \vee b \geqslant a,b; \hfill \\ x \geqslant a,b \Rightarrow x \geqslant a \vee b \hfill \\ {\text{Greatest lower bound: }}a,b \geqslant a \wedge b; \hfill \\ a,b \geqslant x \Rightarrow a \wedge b \geqslant x. \hfill \\ \end{gathered} $$

This special relationship between order properties and algebraic properties for lattices has played an important role in the development of lattice theory. Some of the more significant implications of this relationship will be examined in this paper.

1. A finite partially ordered set is completely determined by the set of covering pairs, namely, those pairs (a,b) for which a > b and there is no x for which a > x > b. It follows that a finite lattice is determined by its set of covering pairs. This leads immediately to the lattice diagram of a finite lattice in which the elements of the lattice are represented by points on a sheet of paper so arranged that for a covering pair (a,b) the point corresponding to a lies above the point corresponding to b and is joined to it by a line segment. Infinite lattices with a sufficiently regular structure can also be diagramed in this manner. From the earliest investigations, lattice diagrams have been an inportant research tool and a useful device for exposition. [5, 21, 25, 31]

2. Associated with any partially ordered set is a lattice (in fact, a complete lattice) obtained as a generalization of Dedekind’s completion by cuts. If A is a subset of the partially ordered set and A* is the set of upper bounds of A and A* is the set of lower bounds of A, then A →(A*)_* is a closure operation on the subsets and the closed subsets under set inclusion are a complete lattice — the normal completion of the partially ordered set. This construction is basic for the study of many embedding problems. When applied to lattices, the completion process preserves the meet and join operations and hence is a lattice embedding. Thus it is not unexpected that the normal completion is useful in the study of purely algebraic problems in lattice theory. [8, 20]

3. Associated with any finite distributive lattice L is the partially ordered set J of its join irreducible elements (q is join irreducible if q = a ∨ b implies q = a or q = b). The order ideals of J form a lattice isomorphic to L. Under this correspondence, many algebraic properties of L are reflected in order- theoretic properties of J. Since the order-theoretic analogues of algebraic problems may be easier to attack, this correspondence frequently provides a promising approach to the study of problems concerning distributive lattices. [2, 11]

4. Much of the structure theory of algebraic systems centers on the study of the lattice of congruence relations. In particular, the lattice of congruence relations on a lattice plays a very important role in exhibiting its lattice structure. As a consequence of being an ordered system the lattice of congruence relations on a lattice is distributive. In turn, the congruence distributivity of lattices underlies many of the most interesting structure theorems for lattices. [12, 14, 17]