Minimal monoids generating varieties with complex subvariety lattices (Q6543260)

A variety of algebras is called \textit{finitely universal} if its lattice of subvarieties contains an isomorphic copy of every finite lattice. While examples of finitely universal varieties of semigroups have been known since the 1970s [\textit{S. Burris} and \textit{E. Nelson}, Proc. Am. Math. Soc. 30, 37--39 (1971; Zbl 0198.34303)], the first finitely universal varieties of monoids were described only relatively recently by \textit{S. V. Gusev} and \textit{E. W. H. Lee} [Bull. Lond. Math. Soc. 52, No. 4, 762--775 (2020; Zbl 1485.20134)], who also proved the existence of \textit{finitely generated} finitely universal varieties of monoids. The minimum order of a finite monoid generating a finitely universal variety was proved to be at least six, but whether this lower bound was attained remained open.\N\NThe author here shows that both the six-element Brandt monoid \N\[\NB_2^1 = \langle a,b \mid aba = a, bab = b, a^2 = b^2 = 0\rangle\N\]\Nand the six-element monoid \N\[\NA_2^1 = \langle a,b \mid aba = a, bab = b, a^2 = 0, b^2 = b\rangle\N\]\Ngenerate a finitely universal variety of monoids. In this way, \(B_2^1\) and \(A_2^1\) are minimal examples of monoids generating a finitely universal variety.\N\NThese results are proved by exhibiting a variety of monoids that is both finitely universal and contained in the variety generated by \(B_2^1\) (which is in turn included in the variety generated by \(A_2^1\)). In addition, the author shows that his variety is included in a join of two Cross varieties, \textit{i.e.}, varieties that are finitely generated, finitely based and contain only finitely many subvarieties. This proves that a join of two Cross varieties can be finitely universal. In fact, the subvariety lattices of the two Cross varieties involved are the \(6\)-element and the \(7\)-element chains, respectively; their join is still finitely universal and in addition contains uncountably many subvarieties.