Finite, integral, and finite-dimensional relation algebras: a brief history

Relation were invented by Tarski and his collaborators in the middle of the twentieth century. The concept of integrality arose naturally early in the history of the subject, and so did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. I recount some results on these topics in chronological order. 1. The calculus of relations and finite-variable logic: 1940–1980. The relative product R|S of two binary relations R and S is defined by R|S := {〈x, y〉 : ∃x(xRz ∧ zSy)}. As a symbol for relative multiplication, the vertical stroke | was introduced by A. N. Whitehead and B. Russell in Principia Mathematica [33, 34, 35]. It is easy to prove that relative multiplication obeys the associative law (1) R|(S|T ) = (R|S)|T. For a proof of the inclusion from left to right, suppose that 〈a, b〉 ∈ R|(S|T ). Then there must be some c such that 〈a, c〉 ∈ R and 〈c, b〉 ∈ S|T . By the latter statement, there must be some d such that 〈c, d〉 ∈ S and 〈d, b〉 ∈ T . From 〈a, c〉 ∈ R and 〈c, d〉 ∈ S we conclude that 〈a, d〉 ∈ R|S, which, combined with 〈d, b〉 ∈ T , gives 〈a, b〉 ∈ (R|S)|T . Note that four objects, namely a, b c, d, are used in the proof. This much was apparent to A. De Morgan[21, 22, 24, 23], C. S. Peirce [26, 27, 28], and E. Schröder [30]. A. Tarski [31, 32] discovered that the reference to four objects is required to prove (1); it can’t be proved with reference to only three objects. To see how to prove this formally, start by restating (1) as a sentence in a first-order language that has only binary relations symbols. The equation R = S between two relations is expressed by the sentence