Combinatorial Rees–sushkevich Varieties That Are Cross, Finitely Generated, or Small

Abstract

A variety is said to be a Rees-Sushkevich variety if it is contained in a periodic variety generated by 0-simple semigroups. Recently, all combinatorial Rees-Sushkevich varieties have been shown to be finitely based. The present paper continues the investigation of these varieties by describing those that are Cross, finitely generated, or small. It is shown that within the lattice of combinatorial Rees-Sushkevich varieties, the set F of finitely generated varieties constitutes an incomplete sublattice and the set S of small varieties constitutes a strict incomplete sublattice of F. Consequently, a combinatorial ReesSushkevich variety is small if and only if it is Cross. An algorithm is also presented that decides if an arbitrarily given finite set of identities defines, within the largest combinatorial Rees-Sushkevich variety, a subvariety that is finitely generated or small. This algorithm has complexity O(nk) where n is the number of identities in and k is the length of the longest word in .2000 Mathematics subject classification: primary 20M07; secondary 03C05, 08B15.

Monoid varieties with extreme properties

“…Many finite semigroups are known to generate varieties of type ℵ 0 . For instance, an abundance of examples can be found from varieties generated by completely 0-simple semigroups [22,31]; the Brandt semigroup B 2 is an example [20] in particular.…”

Finite basis problem for 2-testable monoids

“…By condition ( where σ π is the identity p ( ) u π q ( ) ≈ p ( ) v π q ( ) . By Lemma 2.2(ii), the variety B 1 0 satisfies the identity (12). Therefore it follows from (a) and (b) that the variety B 1 0 satisfies the identity u π ≈ v π , whence by Lemma 2.2(ii), the identity u π ≈ v π does not delete to any identity in (2).…”

“…Then it follows from Lee [8,Theorem 2] that the variety V satisfies the identity ( ) ≈ ( 2 2 ) . It is easy to deduce that the variety V also satisfies the identity (12).…”

“…Let w be any connected, quadratic word. Then the deduction (12) w ≈ w π holds for some peaceful, connected, quadratic word w π .…”

Varieties of monoids with complex lattices of subvarieties

“…Not only is the variety COM nonfinitely generated, it is not contained in any finitely generated variety. The variety H is also non-finitely generated [14], but it is a more interesting example because it is contained in every known example of finitely universal variety generated by a finite semigroup, such as the well-known Brandt semigroup B 2 = a, b | a 2 = b 2 = 0, aba = a, bab = b of order five and even several semigroups of order four [13].…”