Ternary semigroups of morphisms of objects in categories. (English) Zbl 0839.20078

The authors continue the Russian topological school and define ternary semigroups in general categories. Let \({\mathcal K}\) be a category and let \(A\), \(B\) be \({\mathcal K}\)-objects with \(\operatorname{Hom}_{\mathcal K}(A,B)\neq\emptyset\neq\operatorname{Hom}_{\mathcal K}(B,A)\), then \(\text{Sem}_{\mathcal K}(A,B)=(S,\omega)\) is a ternary semigroup where \(S=\operatorname{Hom}_{\mathcal K}(A,B)\times\operatorname{Hom}_{\mathcal K}(B,A)\) and \(\omega((p_0,r_0),(p_1,r_1),(p_2,r_2))=(p_0\circ r_1\circ p_2,r_0\circ p_1\circ r_2)\).
The following obvious fact is proved, if \(F:{\mathcal K}\to{\mathcal L}\) is an isofunctor then \(\text{Sem}_{\mathcal K}(A,B)\) is isomorphic to \(\text{Sem}_{\mathcal L}(FA,FB)\) for any \({\mathcal K}\)-objects \(A\) and \(B\). (If \(F\) is only a full embedding then this fact is also true.) Theorem is illustrated on the category of \((n+1)\)-ary relations and strong homomorphisms and the category of \(n\)-ary power algebras and their homomorphisms.
Reviewer: V.Koubek (Praha)


20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
18B10 Categories of spans/cospans, relations, or partial maps
08A02 Relational systems, laws of composition
08A62 Finitary algebras
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