Lower Level Connections between Representations of Relation Algebras

(abstract) The algebra of all binary relations on a given set is the most important example of a relation algebra (cf. [3]). In this note we will examine the possible isomorphisms within some subclasses of a closely related class (cf. [1] 5.3.2); A is a relation set algebra with base U if its Boolean reduct is a field of sets with unit element 2 U , its universe A contains the identity relation on U and it is closed under the operations The class of all relation set algebras is denoted by Rs. Considering an algebra with a universe which is defined as a collection of some subsets of a given set, the question naturally arises when and to what extent the algebraic structure of this algebra determines its set structure. There are, in fact, cases when the only possible isomorphisms are the trivial ones, the so-called base – isomorphisms (cf. [2] I.3.5-6 on p. 37). Let A be a relation set algebra. We say that a function f defined on A is a base-isomorphism if there is a one-one function g defined on the base of A such that f R = {< gx, gy >:< x, y >∈ R} for any R ∈ A. In order to be able to examine more general cases as well, we introduce some weaker kinds of isomorphisms between set algebras defined in terms of their set structure (cf. [2] 3.1 on pp. 155–156); Let A be a relation set algebra with base U. We say that an isomor-phism f from A onto a relation set algebra is an ext-isomorphism if there is a V ⊆ U such that f R = R ∩ 2 V for any R ∈ A. Further, f is an ext-base-isomorphism if f = e • b for some base-isomorphism b and ext-isomorphism e, while f is said to be a lower-base-isomorphism if f = e −1 1 • b • e 2 for some ext-isomorphisms e 1 , e 2 and base-isomorphism b.