an:01522669
Zbl 0970.20035
Pol??k, Libor
Unification in varieties of completely simple semigroups
EN
Dorninger, D. (ed.) et al., Contributions to general algebra 12. Proceedings of the 58th workshop on general algebra ``58. Arbeitstagung Allgemeine Algebra'', Vienna, Austria, June 3-6, 1999. Klagenfurt: Verlag Johannes Heyn. 337-347 (2000).
2000
a
20M07 20M05 03B35 68T15
varieties of completely simple semigroups; solvability of equations; most general solutions; unification types of varieties
Solving an equation in a variety means solving it in the free object of that vartiety. For a variety \(\mathcal V\) of groups, let \({\mathcal V}CS\) denote the variety of all completely simple semigroups with subgroups in \(\mathcal V\) (completely simple semigroups are treated as unary semigroups). The aim of the paper is to compare the solvability of equations with constants in \({\mathcal V}CS\) and \(\mathcal V\). It turns out that the solvability of an equation in \(n\) variables and \(m\) constants in \({\mathcal V}CS\) translates into the solvability of a disjunction of \(\leq m^{2n}\) equations in \(n\) variables in \(\mathcal V\) (Theorem 1). This is used to show that the unification type of \({\mathcal V}CS\) is finitary (infinitary, nullary) if and only if so is the unification type of \(\mathcal V\) (Theorem 3). The last result, however, does not extend to the unitary unification type: the author exhibits an equation which has 5 different most general solutions in the variety of all completely simple semigroups with Abelian subgroups, while in the variety of Abelian groups, every solvable equation is known to posses a unique most general solution [see \textit{D. Lankford}, \textit{G. Butler} and \textit{B. Brady}, Contemp. Math. 29, 193-199 (1984; Zbl 0555.68065)].
For the entire collection see [Zbl 0942.00022].
Mikhail Volkov (Ekaterinburg)
Zbl 0555.68065