Monoid varieties with extreme properties

Abstract

Finite monoids that generate monoid varieties with uncountably many subvarieties seem rare, and surprisingly, no finite monoid is known to generate a monoid variety with countably infinitely many subvarieties. In the present article, it is shown that there are, nevertheless, many finite monoids with simple descriptions that generate monoid varieties with continuum many subvarieties; these include inherently nonfinitely based finite monoids and all monoids for which xyxy is an isoterm. It follows that the join of two Cross monoid varieties can have a continuum cardinality subvariety lattice that violates the ascending chain condition.Regarding monoid varieties with countably infinitely many subvarieties, the first example of a finite monoid that generates such a variety is exhibited. A complete description of the subvariety lattice of this variety is given. This lattice has width three and contains only finitely based varieties, all except two of which are Cross.

On the ascending and descending chain conditions in the lattice of monoid varieties

“…A useful construction. The following notion was introduced by Perkins [14] and often appeared in the literature (see [4,[6][7][8], for instance). Let W be a set of possibly empty words.…”

“…In [7,Subsection 3.2] two monoid varieties U and W are exhibited such that the subvariety lattices of both varieties are finite, while the lattice L(U ∨ W) is uncountably infinite and does not satisfy the ascending chain condition. Moreover, it follows from the proof of Theorem 3.4 in [7] that L(U∨W) violates the descending chain condition.…”

“…In [7,Subsection 3.2] two monoid varieties U and W are exhibited such that the subvariety lattices of both varieties are finite, while the lattice L(U ∨ W) is uncountably infinite and does not satisfy the ascending chain condition. Moreover, it follows from the proof of Theorem 3.4 in [7] that L(U∨W) violates the descending chain condition. So, the classes of monoid varieties whose subvariety lattices are finite, satisfy the descending chain condition or satisfy the ascending chain condition are not closed with respect to the join of the varieties.…”

Chain varieties of monoids

“…Recently, the situation began to change gradually. The papers [8,9,[12][13][14][15][16] are devoted principally to an examination of identities of monoids but contain also some intermediate results about lattices of varieties. Moreover, the article [9] contains some results about the lattice MON that are of undoubted independent interest.…”

“…We treat this identity element as the empty word and denote it by λ. The following notion was introduced by Perkins [18] and often appeared in the literature (see [8][9][10]12,15], for instance; in [9,Remark 2.4] there is a number of other references). Let W be a set of possibly empty words.…”