The analysis of semigroups of multirelational systems

A finite family of binary relations on a finite set, termed here a relational system, generates a finite semigroup under the operation of relational composition. The relationship between simplifications of the semigroup of a relational system in the form of homomorphic images, and simplifications of the relational system itself is examined. First of all, the list of relational conditions establishing a relationship between a homomorphic image of the semigroup of a relational system and a simplified, or derived, version of that relational system, is reviewed and extended. Then a definition of empirical relationship is introduced (the Correspondence Definition) and it is shown how, in conjunction with a factorization procedure for finite semigroups (P. E. Pattison & W. K. Bartlett, Journal of Mathematical Psychology, 1982, in press), it leads to a systematic and efficient analysis for a relational system. Applications of the procedure to an empirical blockmodel and to a class of simple relational systems are presented.