zbMATH — the first resource for mathematics

Unification in varieties of completely simple semigroups. (English) Zbl 0970.20035
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).
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 D. Lankford, G. Butler and B. Brady, Contemp. Math. 29, 193-199 (1984; Zbl 0555.68065)].
For the entire collection see [Zbl 0942.00022].

20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
03B35 Mechanization of proofs and logical operations
68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)