Varieties generated by unstable involution semigroups with continuum many subvarieties

Lee, Edmond W.H.

Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 44-51.

Avec le temps, on a découvert plusieurs demi-groupes finis engendrant des variétés contenant un ensemble continu de sous-variétés. Toutefois, les demi-groupes involutifs finis qui engendrent des variétés contenant autant de sous-variétés semblent beaucoup plus rares ; en fait, un seul exemple – un demi-groupe inversif d'ordre 165 – a été publié à ce jour. Nous montrons dans le présent article qu'il y a néanmoins beaucoup d'exemples plus petits parmi les demi-groupes involutifs qui sont instables, dans le sens que les variétés qu'ils engendrent contiennent un demi-réseau involutif avec une opération unaire non triviale. Les exemples les plus frappants sont les demi-groupes involutifs finis qui n'ont pas de base finie par essence, le plus petit étant d'ordre 6. Il s'ensuit que le joint de deux variétés engendrées par des demi-groupes involutifs finis et n'ayant qu'un nombre fini de sous-variétés peut contenir un ensemble continu de sous-variétés.

Over the years, several finite semigroups have been found to generate varieties with continuum many subvarieties. However, finite involution semigroups that generate varieties with continuum many subvarieties seem much rarer; in fact, only one example—an inverse semigroup of order 165—has so far been published. Nevertheless, it is shown in the present article that there are many smaller examples among involution semigroups that are unstable in the sense that the varieties they generate contain some involution semilattice with nontrivial unary operation. The most prominent examples are the unstable finite involution semigroups that are inherently non-finitely based, the smallest ones of which are of order six. It follows that the join of two finitely generated varieties of involution semigroups with finitely many subvarieties can contain continuum many subvarieties.

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