A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups (Q2816505)

A semigroup \(S\) is an epigroup if, for each \(x \in S\), some power of \(x\) belongs to a subgroup. If \(x^\omega\) denotes the identity element of that subgroup, then \(x \mapsto \overline{x} = (x^\omega x)^{-1}\) defines a unary operation on \(S\). The class \(\mathfrak{E}\) of epigroups is considered here as as a class of unary semigroups in this way, following the program of \textit{L. N. Shevrin} [Russ. Acad. Sci., Sb., Math. 82, No. 2, 485--512 (1995); translation from Mat. Sb. 185, No. 7, 129--160 (1994; Zbl 0839.20073); Russ. Acad. Sci., Sb., Math. 83, No. 1, 133--154 (1995); translation from Mat. Sb. 185, No. 9, 153--176 (1994; Zbl 0841.20056); in: Structural theory of automata, semigroups, and universal algebra. Proceedings of the NATO Advanced Study Institute, Montreal, Quebec, Canada, July 7--18, 2003. Dordrecht: Kluwer Academic Publishers. 331--380 (2005; Zbl 1090.20032)]; then \(\mathfrak{E}_{\mathrm{fin}}\) denotes the class of finite epigroups (that is, finite semigroups considered in this fashion) and \(\mathfrak{A}_{\mathrm{fin}}\) the class of finite aperiodic semigroups. Although \(\mathfrak{E}\) is not itself a variety, one may investigate the identities it satisfies. The purpose of this paper is to recover the theorem of Zhil'tsov, that the equational theory of \(\mathfrak{E}\) coincides with that of \(\mathfrak{E}_{\mathrm{fin}}\) and is decidable. \textit{I. Yu. Zhil'tsov} announced the results in [Dokl. Math. 62, No. 3, 322--324 (2000; Zbl 1053.20518); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 375, No. 1, 10--12 (2000)], along with an analogue for \(\mathfrak{A}_{\mathrm{fin}}\), found independently by \textit{J. P. McCammond} [Int. J. Algebra Comput. 11, No. 5, 581--625 (2001; Zbl 1026.20037)], but died without his proofs becoming known. Here, the author states that she follows his outline in constructing and proving some results about normal forms, and then follows her own path in completing the proofs. At the end, a rather simple infinite basis of identities is exhibited.NEWLINENEWLINEIndependently, \textit{J. C. Costa} [Discrete Math. Theor. Comput. Sci. 16, No. 1, 159--178 (2014; Zbl 1293.20056)] has proven similar results for \(\mathfrak{E}_{\mathrm{fin}}\), with methods ultimately generalizing instead those of McCammond [loc. cit.].