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The identities of the free product of a pair of two-element monoids. (English) Zbl 1412.20019
The free product in the title is the monoidal one, amalgamating the identities. It is noted that the free product of a monoid containing at least two elements with one containing at least three elements contains a free monoid of rank two and thus satisfies no nontrivial identities. The only cases of interest are then the free products of two two-element semilattices, of a two-element semilattice with a two-element group, and of two two-element groups, denoted respectively by \(J_\infty\), \(K_\infty\) and \(D_\infty\). The main result of the paper states that the identities of \(K_\infty\) are not finitely based. The argument re-uses the technique used by L. M. Shneerson and the author [Semigroup Forum 95, No. 1, 245–250 (2017; Zbl 1375.20060)] to show that the analogue holds for \(J_\infty\), which is just the semigroup free product of two trivial semigroups, having an adjoined identity. The author observes that a modification of this technique also yields the analogue for \(D_\infty\), the infinite dihedral group, although this may also be deduced from general results. Finally, it is observed that no two of the three monoids above satisfy the same sets of identities.
MSC:
20M10 General structure theory for semigroups
20M05 Free semigroups, generators and relations, word problems
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