The Semigroups B\u3csub\u3e2\u3c/sub\u3e and B\u3csub\u3e0\u3c/sub\u3e 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 behavior 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 �. For example, explicit bases of identities are found for the varieties generated by B0 and B2

A minimal nonfinitely based semigroup whose variety is polynomially recognizable

We exhibit a 6-element semigroup that has no finite identity basis but nevertheless generates a variety whose finite membership problem admits a polynomial algorithm.Comment: 16 pages, 3 figure

The algebra of adjacency patterns: Rees matrix semigroups with reversion

We establish a surprisingly close relationship between universal Horn classes of directed graphs and varieties generated by so-called adjacency semigroups which are Rees matrix semigroups over the trivial group with the unary operation of reversion. In particular, the lattice of subvarieties of the variety generated by adjacency semigroups that are regular unary semigroups is essentially the same as the lattice of universal Horn classes of reflexive directed graphs. A number of examples follow, including a limit variety of regular unary semigroups and finite unary semigroups with NP-hard variety membership problems.Comment: 30 pages, 9 figure

A new example of a limit variety of monoids

A variety of universal algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. Until recently, only two explicit examples of limit varieties of monoids, constructed by Jackson, were known. Recently Zhang and Luo found the third example of such a variety. In our work, one more example of a limit variety of monoids is given.Comment: 16 page

Equidivisible pseudovarieties of semigroups

We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Besides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal'cev product on the left by the pseudovariety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion