zbMATH — the first resource for mathematics

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.

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