Free inverse semigroups are not finitely presentable

Abstract

In the memory of Professor A. Kert&zFree inverse semigroups became a subject of intense studies in the last few years. Their existence was proved long ago: as algebras with two operations (binary multiplication and unary involution) inverse semigroups may be characterized by a finite system of identities, i.e. they form a variety of algebras [10]. Therefore, free inverse semigroups do exist.A construction of a free algebra in a variety of algebras (as a quotient algebra of an absolutely free word algebra) is well known. Free inverse semigroups in such a form were considered by V. V. VAO_' ,,~ER [14] who found certain properties of such semigroups. A monogenic free inverse semigroup (i.e. a free inverse semigroup with one generator) was described by L. M. GLUSKIN [2]. Later this semigroup was described by H. E. SCHEIBLICH in a slightly different form [8]. The most essential progress in this direction was made in a paper [9] by H. E. SCHEIBLlCrI who described arbitrary free inverse semigroups. A relevant paper [1] by C. EBERHART and J. SELDEN should be mentioned. There are papers on some special properties of free inverse semigroups. N. R. REILLV described free inverse subsemigroups of free inverse semigroups [7], results in this direction were obtained also by W. D. MUNN and L. O'CARROLL.Let ff'-'cx denote the free inverse semigroup with the set X of free generators. A monogenic free inverse semigroup will be denoted ffd 1. Time and then we will write ~r instead of YJx. We do not consider ~Jo, a one-element inverse semigroup.This paper contains two main results. The first one coincides with the title, the second consists in a description of flee inverse semigroups (if a free inverse semigroup is presented as a quotient algebra of a free involuted semigroup, then each element of ffJ is a c]ass of equivalent words, we give a canonical form of the words). Certain corollaries witb~ properties of free inverse semigroups follow.All results of the paper were reported by the author at a meeting of the seminnar "Semigroups" in the Saratov State University on October 21, ~971.THEOREM I. Free inverse semigroups are not finitely presentable either as semigroups or as involuted semigroups.The proof of the theorem is subdivided in a series of lemmas.LEMMA 1. A semigroup F generated by two elements u and v satisfying the infinite list of defining relations: I) uvu=u, vuv=v; Am,,) umv"+"u"=v"u"+"v'~ for all natural m and n, is a free inverse semigroup. Ac~a Mathematica Academiae Scientiarum Hungaricae 26, 197~

Presentations for Subgroups of Monoids

“…A question then arises as to the relationship between monoid presentations and inverse monoid presentations for a given inverse monoid S. It is easy to see that every monoid presentation for S is also an inverse monoid presentation for S. By way of contrast, Schein [22] proved that the free monogenic inverse monoid is not finitely presented as a monoid.…”

“…Finally, the relation (26) can be eliminated by using (22) Proof. The "if" part follows from Corollary 4.7 and Theorem 4.1, and the "only if" part follows from the fact that every monoid presentation for S is also an inverse monoid presentation for S.…”

Presentations of Inverse Semigroups, Their Kernels and Extensions

“…Indeed, in [40] it was shown that the free inverse semigroup (of any rank) is not finitely presented as a semigroup. Throughout this paper we shall work with inverse semigroups, inverse semigroup presentations, and by finitely presented we shall always mean finitely presented as an inverse semigroup.…”

On finite generation and presentability of Schützenberger products

“…Indeed, if S is inverse and 5 = Mon{X\R) then, by adding the standard inverse monoid relations (see (T2) below), S = Im{X\R). The converse does not hold in general: in [10] it is shown that the free inverse monoid is not finitely presented as a monoid (this result also appeared as Theorem IX.4.7 of [7]). …”