Compositional Homomorphisms of Relational Structures (modeled as Multialgebras)

The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented as a set-valued function). The rst, main, result is that, under reasonable restrictions on the form of the deenition of homomorphism, there are exactly nine compositional homomorphisms of multialgebras. Then the comparison of the obtained categories with respect to the existence of nite limits and co-limits reveals two of them to be nitely complete and co-complete. Without claiming that compositionality and categorical properties are the only possible criteria for selecting a deenition of homomorphism, we nevertheless suggest that, for many purposes, these criteria actually might be acceptable. For such cases, the paper gives an overview of the available alternatives and a clear indication of their advantages and disadvantages. 1 Background and motivation In the study of universal algebra, the central place occupies the pair of \dual" notions of congruence and homomorphism: every congruence on an algebra induces a homomorphism into a quotient and every homomorphism induces a congruence on the source algebra. Categorical approach attempts to express all (internal) properties of algebras in (external) terms of homomorphisms. When passing to relational structures or power set structures, however, the close correspondence of these internal and external aspects seems to get lost. The most common, and natural, generalisation of the deenition of homomor-phism to relational structures says: Deenition 1.1 A set function : A ! B, 1 where both sets are equipped with respective relations R A A n and R B B n , is a (weak) homomorphism ii hx 1 Underlying sets will be used to indicate the \bare, unstructured sets" as opposed to power sets or other sets with structure. For the moment, one may ignorte this notational convention.