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Induced isomorphisms of certain ternary semigroups. (English) Zbl 0840.20072

From the authors’ abstract: If \(X_1\), \(Y_1\) are relational structures of the same type, then the set of all ordered pairs \((p,q)\) constitutes a ternary semigroup with a naturally defined operation where \(p\) denotes a homomorphism of \(X_1\) into \(Y_1\) and \(q\) is a homomorphism of \(Y_1\) into \(X_1\). If \(f_1\) is an isomorphism of \(X_1\) onto a relational structure \(X_2\) and \(f_2\) an isomorphism of \(Y_1\) onto a relational structure \(Y_2\), then the ordered pair \((f_1,f_2)\) of isomorphisms defines an isomorphism of the ternary semigroup defined on the basis of \(X_1\) and \(Y_1\) onto the ternary semigroup defined on the basis of \(X_2\) and \(Y_2\); this isomorphism is said to be induced. We prove that there exist isomorphisms of ternary semigroups defined by pairs of relational structures that are not induced and formulate a criterion recognizing induced isomorphisms.

MSC:

20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
08A02 Relational systems, laws of composition
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