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It gives me great pleasure to present this brief commentary on some of T. P. Speed’s papers on algebra. It may come as a surprise to many of Speed’s colleagues to know that his 1968 PhD thesis was entitled Some Topics in the Theory of Distributive Lattices. Moreover, of his first 15 papers only one was in probability theory with the remainder in algebra. Nevertheless, this fruitful excursion into algebra has its roots in the foundations of probability theory. In the introduction to his PhD thesis, Speed writes:

In July 1965, the author began to look at the lattices associated with intuitionistic logic which are called variously – relatively pseudo-complemented, brouwerian or implicative lattices. This was under the direction of Professor P. D. Finch and aimed towards defining probability measures over these lattices. It was hoped that a probability theory could be developed for the intuitionistic viewpoint similar to the Kolmogorov one for classical logic.

Speed never returned to the search for an intuitionistic probability theory for, as he says later in the introduction to his thesis, he became “sold on distributive lattices”. In the summer of 1968–1969, between my third and honours years, I spent three months on a Monash University Graduate Assistantship during which I read Speed’s PhD thesis. By the end of that summer I was also sold on distributive lattices and have been ever since [2].

Between 1969 and 1974, Speed published 17 papers on a range of algebraic topics: distributive lattices, including their topological representation (9), Baer rings (3), Stone lattices (2), semigroups (2), and ℓ-groups (1). In the commentary below, I will discuss five of these papers. Only one of these papers, the first discussed, comes from Speed’s thesis.

Distributive lattices in general

Most of Speed’s work on distributive lattices revolves around the role of particular sorts of prime ideals, with an emphasis on minimal prime ideals. In this section, we will look at two of the seven papers that fall into this category, namely, On rings of sets [10] and On rings of sets. II. Zero-sets [16].

In the first of these papers, Speed provides a unified approach to a number of representations of distributive lattices as rings of sets, that is, as lattices of subsets of some set in which the operations are set-theoretic union and intersection. Each of these characterisations was originally given in terms of the existence of enough elements of a special form, and their proofs looked quite different. Given cardinals \(\mathfrak{m}\) and \(\mathfrak{n}\), a lattice L is called \((\mathfrak{m},\mathfrak{n})\)-complete if it is closed under the operations of least upper bound and greatest lower bound of sets of at most \(\mathfrak{m}\) and \(\mathfrak{n}\) elements, respectively. An \((\mathfrak{m},\mathfrak{n})\)-complete lattice of sets is an \((\mathfrak{m},\mathfrak{n})\)-ring of sets if \(\mathfrak{m}\)-ary least upper bounds and \(\mathfrak{n}\)-ary greatest lower bounds are given by set union and intersection, respectively. For example, the open sets of a topological space form an \((\mathfrak{m},2)\)-ring of sets for every cardinal \(\mathfrak{m}\). Speed introduces \(\mathfrak{n}\)-prime \(\mathfrak{m}\)-ideals and employs them to give natural necessary and sufficient conditions for an \((\mathfrak{m},\mathfrak{n})\)-complete lattice to be isomorphic to an \((\mathfrak{m},\mathfrak{n})\)-ring of sets. As Speed remarks in the introduction to the paper, It is interesting to note that the elementary methods used in representing distributive lattices carry over completely and yield all these results, although this is hardly obvious when one considers special elements of the lattice.

In On rings of sets. II. Zero-sets [16], Speed turns his attention to an important example of (2, ω)-rings of sets, the lattice Z(X) of zero-sets of continuous real-valued functions on a topological space X. The paper, which is deeper and somewhat more technical than the first, includes lattice-theoretic characterisations of Z(X) in two important cases, when X is compact (Theorem 4.1) and when X is an arbitrary topological space (Theorem 5.9). In both cases, the characterisations involve minimal prime ideals. Along the way he proves a result (Theorem 3.1) that very nicely generalises Urysohn’s Lemma for normal topological spaces and the fact that, in a completely regular space, disjoint zero-sets can be separated by a continuous function.

Distributive lattices—Priestley duality

About the same time that Speed was writing his PhD thesis at Monash University, H. A. Priestley was writing her DPhil at the University of Oxford. Speed was amongst the first to realise the importance of the new duality for bounded distributive lattices that Priestley established in her thesis (see Priestley [8, 9] and Davey and Priestley [2]).

In On the order of prime ideals [13], Speed addresses the question, raised by Chen and Grätzer [1], of characterising representable ordered sets, that is, ordered sets that arise as the ordered set of prime ideals of a bounded distributive lattice. By using Birkhoff’s duality between finite distributive lattices and finite ordered sets, he shows that an ordered set is representable if and only if it is the inverse limit of an inverse system of finite ordered sets. Speed observes that, when combined with deep results of Hochster [5], this tells us that an ordered set is isomorphic to the ordered set of prime ideals of a commutative ring with unit if and only if it is isomorphic to an inverse limit of finite ordered sets. This cross fertilisation in Speed’s work between commutative rings with unit and bounded distributive lattices will arise again in Section 1.

Soon after writing Speed [13], Speed became aware of Priestley’s results. He quickly realised that, since an inverse limit of finite sets is endowed with a natural compact topology, his characterisation of representable ordered sets could be lifted to a characterisation of compact totally order-disconnected spaces, the ordered topological spaces that arise in Priestley duality (and are now referred to simply as Priestley spaces). In Profinite posets [12], he proved that an ordered topological space is a Priestley space if and only if it is isomorphic, both order theoretically and topologically, to an inverse limit of finite discretely topologised ordered sets.

Baer rings

Speed’s PhD thesis was strongly influenced by the seminal paper Minimal prime ideals in commutative semigroups [6]. He took ideas from Kist’s paper and reinterpreted them in the context of distributive lattices. Speed saw that there was some informal connection between the commutative Baer rings introduced and studied in Kist [6] and Stone lattices, a class of distributive lattices introduced by Grätzer and Schmidt [4]. A commutative ring R is a Baer ring if, for every element a ∈ R, the annihilator \(\mathrm{ann}(a) :=\{\, x \in R\mid xa = 0\,\}\) is a principal ideal generated by a (necessarily unique) idempotent a  ∗ . A bounded distributive lattice L is a Stone lattice if, for every element a ∈ L, the annihilator \(\mathrm{ann}(a) :=\{\, x \in L\mid x \wedge a = 0\,\}\) is a principal ideal generated by an element a  ∗ , and in addition the equation a  ∗  ∨ a  ∗ ∗  = 1 is satisfied. While quite different looking, the requirements that a  ∗  be an idempotent, in the ring case, and the identity a  ∗  ∨ a  ∗ ∗  = 1, in the lattice case, guarantee that the elements a  ∗  form a Boolean algebra and correspond precisely to the direct product factorisations of the ring or lattice.

While the proofs will typically be quite different, it is often true that a result about Baer rings will translate to a corresponding result about Stone lattices and vice versa. For example:

  1. (i)

    Grätzer [3] proved that Stone lattices form an equational class; Speed and Evans [17] proved that Baer rings also form an equational class. (In both cases,  ∗  is added as an additional unary operation.)

  2. (ii)

    Grätzer and Schmidt [4] proved that, in a Stone lattice, each prime ideal contains a unique minimal prime ideal; Kist [6] proved that precisely the same condition holds in a Baer ring.

In separate papers on Stone lattices [11] and Baer rings [14], Speed proves that there are broad classes of distributive lattices and rings, respectively, within which Stone lattices and Baer rings are characterised by the property that each prime ideal contains a unique minimal prime ideal.

In his third and final paper on Baer rings [15], Speed considers the question of embedding a commutative semiprime ring R into a Baer ring B. Two such embeddings had already been given: the first by Kist [6] and the second by Mewborn [7]. In both cases, the Baer ring B was constructed as a ring of global sections of a sheaf over a Boolean space. Speed shows that, in fact, there is a hierarchy of Baer extensions of R, the smallest being Kist’s and the largest Mewborn’s. Moreover, he is able to replace the sheaf-theoretic construction with a purely algebraic one similar in nature to one that had been used previously in the theory of lattice-ordered groups. The underlying lattice of a lattice-ordered group is distributive, so again we see Speed’s fruitful use of the interplay between rings and distributive lattices.