Relation algebras and Schröder categories (Q1105596)

This paper begins with the definition of relation algebras \((A,+,\cdot,\quad -,0,1,;,^{\sqcup},')\) and a recall of some of their properties. A Schröder category, as defined by Olivier and Serrato, is a category \({\mathfrak C}\) such that for every i,j\(\in Ob {\mathfrak C}\), the set \({\mathfrak C}(i,j)\) is a Boolean algebra and there is an involution \(^{\sqcup}: {\mathfrak C}(i,j)\to {\mathfrak C}(j,i)\); besides, the composition of morphism and \(^{\sqcup}\) satisfies certain supplementary conditions. The author proves that if I is any finite set of objects of a Schröder category \({\mathfrak C}\), then the direct product of the Boolean algebras \({\mathfrak C}(i,j)\), \(i,j\in I\), can be made into a relation algebra. Then, a natural concept of tensor product of Boolean algebras is introduced which turns out to be the same as the free product. The next section deals with Boolean modules. Further, given a system of simple relation algebras \({\mathfrak A}_ i\), \(i\in I\), the author constructs from them in a canonical way a Schröder category \({\mathfrak C}\) whose objects are the members of I in such a way that \({\mathfrak C}(i,i)={\mathfrak A}_ i\), and assuming that I is finite, the semi-product of the algebras \({\mathfrak A}_ i\) is defined as a certain algebra \({\mathfrak A}\) of matrices a with entries \(a(i,j)\in {\mathfrak C}(i,j)\); the algebras \({\mathfrak A}_ i\) determine \({\mathfrak A}\) up to isomorphism. The last sections study in some detail the equivalence elements of a relation algebra (i.e., the elements u such that \(u;u\leq u\) and \(u^{\sqcup}=u)\) and the relation algebras that are generated by an equivalence element. Every such algebra is finite and representable.