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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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