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Inner semigroup extension of certain semigroup amalgams. (English. Russian original) Zbl 0839.20072
Russ. Math. 37, No. 11, 18-24 (1993); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 11, 20-26 (1993).
The problem is whether a given partial binary operation $$\xi$$ in a set $$M$$ (the author says “a partial action”) can be extended to an everywhere defined associative operation $$\xi_1$$ in the same set, i.e. $$\xi\subset \xi_1$$ (so called inner semigroup extension because no new elements are added). The author considers this problem for a partial groupoid $$(S, \tau)$$ which is an amalgam of semigroups in a special form. It means that there are a semigroup $$(S, \alpha)$$ and a family of semigroups $$(B_i, \beta_i)$$, $$i\in I$$, such that $$S= A\cup (\bigcup_{i\in I} B_i)$$, $$\tau$$ is the set theoretical union of the partial operations $$\alpha$$ and $$\beta_i$$, and these ones are the restrictions of $$\tau$$ to the respective sets. It is assumed that $$\tau$$ is weakly associative (if $$(ab)c$$, $$a(bc)$$ both exist then they coincide), moverover $$A$$ is a weak bilateral ideal of $$(S, \tau)$$, and there is a system of semigroup homomorphisms $$\varphi_i: B_i\to A$$ which coincide pairwise on the intersections of their domains and are identical on $$A\cap B_i$$, $$i\in I$$. In this case, a total operation $$\theta$$ on $$S$$ is constructed and a necessary and sufficient condition is given for $$\theta$$ to be associative. As a consequence of this result, let one be mentioned: A semigroup amalgam with properties assumed above, having $$i\neq j\Rightarrow B_i\cap B_j \subset A$$, for all $$i,j\in I$$, posseses an inner semigroup extension.

##### MSC:
 20M10 General structure theory for semigroups 08A55 Partial algebras 20M05 Free semigroups, generators and relations, word problems 20M15 Mappings of semigroups