Minimal monoids generating varieties with complex subvariety lattices

Abstract

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. We show that the 6-element Brandt monoid generates a finitely universal variety of monoids and, by the previous results, it is the smallest generator for a monoid variety with this property. It is also deduced that the join of two Cross varieties of monoids can be finitely universal. In particular, we exhibit a finitely universal variety of monoids with uncountably many subvarieties which is the join of two Cross varieties of monoids whose lattices of subvarieties are the 6-element and the 7-element chains, respectively.

22 pages. In version 3, Remark 4 is updated

References in corpus (4)

• The lattice of varieties of monoids
• From $A$ to $B$ to $Z$
• Varieties of monoids with complex lattices of subvarieties
• Two weaker variants of congruence permutability for monoid varieties