A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based. MSC: 20M07 Brought to you by | Nova Southeastern University

more »

... d Download Date | 2/27/17 11:46 PM Finite basis problem for 2-testable monoids satisfy any identity formed by a pair of words that begin with the same letter, end with the same letter, and share the same set of factors of length two. The variety A 2 also coincides with the variety generated by all aperiodic 0-simple semigroups [5] and is essential in the recent discovery and description of a new infinite series of limit varieties [16] . The semigroup A 2 is also an important example that is related to semigroups with very extreme and contrasting equational properties. By the early 1980s, Trahtman [22, 24] had proven that the semigroup A 2 is finitely based by the identities Recently, the semigroup A 2 was shown to satisfy the stronger property of being hereditarily finitely based [9] , that is, every semigroup in the variety A 2 is finitely based. On the other hand, the semigroup A 2 can be used to construct non-finitely based semigroups. Volkov [26] demonstrated that the direct product of the semigroup A 2 with any finite group is non-finitely based. Trahtman [23] proved that the monoid A 1 2 obtained from A 2 by adjoining a unit element is non-finitely based, and Sapir [19] even proved that A 1 2 is inherently non-finitely based in the sense that any locally finite variety containing it is non-finitely based. Brought to you by | Most of the notation and background material of this article are given in this section. Refer to the monograph of Burris and Sankappanavar [2] for more information on universal algebra.