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Homomorphisms and strong homomorphisms of relational structures. (English) Zbl 0909.08001
Let \(A\) be a nonvoid set, \(n\geq 2\) an integer. The ordered pair \((A,r)\) where \(r\subseteq A^n = A \times \ldots \times A\) (\(n\)-times) is called an \(n\)-ary relational structure. Let \((A,r)\), \((A',r')\) be \(n\)-ary relational structures, \(h:A\rightarrow A'\) a mapping. Then \(h\) is called a homomorphism of \((A,r)\) into \((A',r')\) whenever \((x_1,\ldots ,x_n) \in r\) implies \((h(x_1),\ldots ,h(x_n))\in r'\) for any \(x_1,\ldots ,x_n \in A\); \(h\) is called a strong homomorphism of \((A,r)\) into \((A',r')\) whenever for any \(x_1,\ldots ,x_{n-1} \in A\) and any \(x'_n \in A'\) the condition \((h(x_1),\ldots ,h(x_{n-1}),x'_n)\in r'\) holds if and only if there is an element \(x_n \in A\) s.t. \((x_1,\ldots ,x_{n-1},x_n) \in r\) and \(h(x_n) = x'_n\). The present paper reduces the construction of all homomorphisms of an \(n\)-ary relational structure into another structure of the same type to the construction of all strong homomorphisms between suitable \(n\)-ary relational structures. Thus, the construction investigated in the paper may be successively reduced to the construction of suitable homomorphisms of a mono-unary algebra into another algebra of the same type. All homomorphisms of a mono-unary algebra into another one (a problem of Professor O. Borůvka) were already found in the author’s previous papers.
Reviewer: J.Duda (Brno)

MSC:
08A02 Relational systems, laws of composition
08A60 Unary algebras
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