The Semigroups B2 and B0 are Inherently nonfinitely Based, as restriction Semigroups

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {·,−1 }, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups – essentially, ‘forgetting’ the inverse operation x→ x−1 and retaining the induced operations x → x = xx−1 and x → x∗ = x−1x – it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behaviour is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable ‘modulo monoids’. These results are consequences of – and discovered as a result of – an analysis of varieties of ‘strict’ restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of ‘completely r-semisimple’ restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation D. For example, explicit bases of identities are found for the varieties generated by B0 and B2. Completely 0-simple semigroups have played a central role in semigroup theory from the very beginnings of its history. So it is naturally of great interest to study the varieties that they generate, together with their subvarieties. These so-called Rees-Sushkevich varieties have received considerable attention in recent years. (For example [19, 21, 22, 27, 28].) Regarded instead as unary semigroups, the inverse semigroups that are completely 0-simple – the Brandt semigroups – likewise generate varieties of inverse semigroups, though in this context the entire situation was clarified some decades ago [26, XII.4]. We take here an intermediate path that quite naturally lies in the realm of varieties of restriction semigroups, which are biunary semigroups in the signature (·,+ ,∗ ). Although such semigroups are known to arise in several contexts (for a survey, see [11]), for the purposes of this paper we need only view them as follows. Any inverse semigroup (S, ·,−1 ) may be regarded as a restriction semigroup under the induced operations x → x+ = xx−1 and x → x∗ = x−1x, forgetting the inverse operation entirely. The restriction semigroups are the members of the