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On a power of relational structures

1985; Springer Nature; Volume: 35; Issue: 1 Linguagem: Inglês

10.21136/cmj.1985.102006

1572-9141

Vítězslav Novák,

Semantic Web and Ontologies

The aim of this paper is to define direct operations and an operation of a power for relational structures and to prove their properties.In particular, a power satisfies the expected rules with the exception of (G")^ ^ G'^**.We derive sufficient con ditions for the vahdity of that law.1. Let / 7^ 0 be a set, let n-, be a positive integer for any i e I.A family (n^; i e /) will be called a type.The types {n^\ iel), {m^je j) are similar iff there exists a bijection cp: I ~-> J such that m^(i) = rii for all / e /.2. Definition.Let G 7^ 0 be a set, let (п^; i e /) be a type.Let C^ be an ?г-агу relation on the set G for any i e /, i.e.C^ я G"\ Then G = (G, (С,; / e i)) is called a relational structure of type (n^; / e/).If G" = (G, (Cj; / G/)) is a relational structure, then the set G is called a carrier of G and C^ are called relations of G. Sometimes we denote by ^i{G) the i^^ relation of G = (G, (Cil iel)), i.e. ^,(G) = C,.Two relational structures G = (G, (C^; / G/)) and H = (Я, (i^^; j e J)) of types (Wj-; iel) and {mj\jeJ), respectively, are called similar iff their types (n^; ie/) and (m^; j e J) are similar.