Review of Introduction to Lattices and Order ( B . a . Davey

The idea that a set may come equipped with a natural ordering on its elements is so basic as to pass by unnoticed by most. However, this belies a wonderful opportunity for unifying the study of many disparate examples of these creatures. Historically, the study of order has led to a great unification of results in algebra and logic. More recently, it has infused into theoretical computer science, particularly into programming language semantics. Given two elements, x and y, in some partially ordered set, or poset, P , it is often the case that they have both a least upper bound, lub{x, y}, and a greatest lower bound, glb{x, y}. An example of such a poset is provided by the natural numbers under their usual order. However, this property certainly does not hold for every poset. For instance, suppose that the underlying set of P consists of those finite subsets of N that have even cardinality. This becomes a poset by saying that x is less than or equal to y just in case x is a subset of y. Now, suppose that x = {1, 2} and y = {2, 3}. Any upper bound of both x and y has to contain x∪y = {1, 2, 3}. Necessarily, a least upper bound would have cardinality 4. However, there is no least such set containing {1, 2, 3}. If we are in the happy situation that any two elements of our poset P have both a least upper bound and a greatest lower bound, then we say that P is a lattice. If P is a lattice and x, y ∈ P , then let us save on effort by writing glb{x, y} as x ∨ y and lub{x, y} as x ∧ y. The operator ∨ is called join and the operator ∧ is called meet. Playing around with these operators a bit, one may notice some identities starting to crop up. Amongst those identities that hold for elements a, b, c ∈ P , we find the following: a ∧ a = a a ∨ a = a a ∧ b = b ∧ a a ∨ b = b ∨ a a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∨ (b ∨ c) = (a ∨ b) ∨ c a ∧ (a ∨ b) = a a ∨ (a ∧ b) = a In fact, the above equations are sufficient to define P by making the observation that a ≤ b if and only if a = a ∧ b. So, we have two ways of looking at a lattice. That is, we can see it as either a special sort of poset or as a set with two binary operators satisfying the above equalities. This brings us to the book under review, which sets out to provide a thorough introduction to both points of view. Examples of lattices abound in theoretical computer science and, indeed, the book makes no bones about dipping into computational applications on a regular basis. Let’s see what’s between the covers, shall we?