THE SEMIGROUPS B₂AND B₀ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

Abstract

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 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

Varieties of restriction semigroups and varieties of categories

“…We first introduce restriction semigroups more formally, along with their basic properties and related definitions. In addition to referring the reader to the general citations [4,5], especially for historical background, we do likewise to the 'prequels' [7,8] for more concise general background and for more information about strict restriction semigroups and their varieties. Some of the material below is repeated, for convenience, from the latter.…”

“…In one sense this paper is a continuation of the study of the lattice of varieties of restriction semigroups, initiated by the author in [7] and narrowed in [8], where attention was focused on characterizations of the subvariety B of strict restriction semigroups and the subvarieties B 2 and B 0 generated respectively by B 2 and B 0 . Here B 2 is the five-element Brandt semigroup and B 0 is the restriction subsemigroup obtained by deleting one of its nonidempotents.…”

“…It was shown in [8] that the lattice L(B 2 ) consists of the sequence of coverings B 2 B 0 S T, where S denotes the variety of semilattices (considered as restriction semigroups). This provides a second decomposition of L(B) into four 'layers', the top one being [B 2 , B], the next [B 0 , B 0 M], then [S, SM] and [T, M], the last two forming the decomposition of the lattice of varieties of semilattices of monoids determined in [7].…”

“…The techniques that we develop to study the interval [B 2 , B] also apply to the interval [B 0 , B 0 M], since the restriction semigroups belonging to B 0 M were characterized in [8] in a manner that translates easily into category-theoretic terms. The proof of the theorem of Tilson cited above enables the analogous covering B 0 M ∩ mon(T) B 0 to be proved.…”

“…The tool behind the varietal connection is the identification of categories with 'primitive' restriction semigroups. While this identification has long been known (a very special case of general work of Lawson [9]), what to us is most important is identification of the connected non-monoid categories with the 'completely 0-r-simple' restriction semigroups introduced by the author in [8]. Based on the concept of 'r-ideal', it was shown there that the strict restriction semigroups are precisely the subdirect products of such semigroups (and monoids).…”