The free Archimedean epigroups of finite degree (Q1281618)

Any semigroup \(S\) in which some power of each element lies in a subgroup may be endowed with a natural unary operation. Namely, for every \(x\in S\), there is a unique maximal subgroup \(H_e\) (with the identity element \(e\)) containing all but finitely many powers of \(x\), and the product \(xe\) is known to belong to \(H_e\). This gives rise to the operation \(x\mapsto\overline x\) where \(\overline x\) denotes the inverse of \(xe\) in the group \(H_e\). \textit{L. N. Shevrin} [Mat. Sb. 185, No. 7, 129-160 (1994; Zbl 0839.20073)] suggested the name ``epigroup'' for \(S\) equipped with this unary operation. An Archimedean epigroup \(S\) is an ideal extension of a completely simple ideal by a nil-semigroup. If the nil-semigroup is nilpotent of degree \(\leq m\) (that is, it satisfies \(x_1\cdots x_m=0\)), then \(S\) is said to be Archimedean of degree \(\leq m\). For every \(m\) and for every group variety \(\mathcal H\), the class \({\mathcal A}_m({\mathcal H})\) of all Archimedean epigroups of degree \(\leq m\) whose maximal subgroups belong to \(\mathcal H\) constitutes a variety of epigroups as algebras of the type \((2,1)\). The author exhibits a model for the free epigroup in the variety \({\mathcal A}_m({\mathcal H})\) (Theorem 2.1) and solves the word problem in this free epigroup modulo the word problem in the free group of the variety \(\mathcal H\) (Corollary 3.5). He also shows that the maximal subgroups of the \({\mathcal A}_m({\mathcal H})\)-free epigroup with \(k\) generators are \(\mathcal H\)-free groups with \(k^m(k-1)+1\) generators (Corollary 3.1) while the kernel of the epigroup is not free in the corresponding variety of completely simple semigroups (Corollary 3.3). The author mentions that an alternative description of the free epigroup in the variety of all Archimedean epigroups of degree \(\leq m\) can be extracted from some results by \textit{J. Almeida} and \textit{A. Azevedo} [Port. Math. 50, No. 1, 35-61 (1993; Zbl 0802.20049)] and discusses the differences between the two approaches.