A note on the finite basis and finite rank properties for pseudovarieties of semigroups

Abstract

The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based. In this note, we establish a result on nilpotent pseudovarieties which yields many similar examples in the realm of pseudovarieties of semigroups.

Towards a pseudoequational proof theory

“…In case one of the first two conditions holds, the mapping from S onto the rectangular band T = A/ρ 1 × A/ρ 2 that maps each triple (x, w, y) ∈ S to (x/ρ 1 , y/ρ 2 ) is a continuous homomorphism onto a semigroup from CS that distinguishes u and v. Moreover, its kernel congruence is contained in ρ, which implies that T ∈ Σ , contradicting the assumption that Σ satisfies u = v. Hence, we may assume that (a, c) ∈ ρ 1 and (b, d) ∈ ρ 2 , so that g ω−1 h does not belong to N ρ . As in the proof of Theorem 8.1, we deduce that there is a clopen normal subgroup K of Ω X G such that N ρ ⊆ K and g ω−1 h / ∈ K. Since K contains N ρ , the triple (ρ 1 , ρ 2 , K) still satisfies the analogues of conditions (12) and (13). Hence, by Theorem 9.1, it defines a congruenceρ on S. Since K has finite index in Ω X G, the congruenceρ has finite index in S. The reader may easily verify that, since ρ is a closed congruence on S and K is a closed subgroup of Ω X G, the congruenceρ is still closed, whence the natural mapping S → S/ρ is a continuous homomorphism.…”

“…ρ 2 ) is an equivalence relation on the set I (resp. Λ) and N ρ is a normal subgroup of G, satisfying properties (12) and (13), the relation ρ τ = (i, g, λ), (j, h, µ) ∈ S × S : i ρ 1 j, λ ρ 2 µ, gN ρ = hN ρ is a congruence on S and every congruence on S is of this form.…”

“…Some sort of topological closure operator seems to be required. While such an operator was also proposed by the first author (see [2, Section 3.8]), it is very hard to handle and only one instance of its application has been found so far [12].…”