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Categories of Relational Structures
, 1998
"... . The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures  viewed as multialgebras  reveals the one with the most desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to s ..."
Abstract

Cited by 11 (3 self)
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. The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures  viewed as multialgebras  reveals the one with the most desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to sets (thus being relations). Finally, we indicate some consequences of our results for partial algebras which are special case of multialgebras. 1 Introduction 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, however, the close correspondence of these internal and external aspects seems to get lost. The most common, and natural, gene...
Compositional Homomorphisms of Relational Structures (Modeled As Multialgebras)
, 2003
"... The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented as a setvalued function). The first, main, result is that, under reasonable restrictions on the form of the definition of homomorphism, there ..."
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The paper attempts a systematic study of homomorphisms of relational structures. Such structures are modeled as multialgebras (i.e., relation is represented as a setvalued function). The first, main, result is that, under reasonable restrictions on the form of the definition of homomorphism, there are exactly nine compositional homomorphisms of multialgebras. Then the comparison of the obtained categories with respect to the existence of finite limits and colimits reveals two of them to be finitely complete and cocomplete. Without claiming that compositionality and categorical properties are the only possible criteria for selecting a definition 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.