Congruence Networks on Regular Semigroups Associated with 𝒱-Relation

Abstract

In this article, the congruence lattice on a regular semigroup is studied through 𝒯𝒱-networks. Let S be a regular semigroup and 𝒞(S) the congruence lattice of S. The 𝒦 [𝒯]-relation identifies 2 congruences on S if they have the same kernel [trace]. For ρ, θ ∈ 𝒞(S), , . For every ρ ∈ 𝒞(S) there exist a greatest congruence ρT (respectively, ρK, ρU, and ρV) and a smallest congruence ρt (respectively, ρk, ρu, and ρv) on S in the same 𝒯 (respectively, 𝒦, 𝒰, and 𝒱)-class as ρ. It is shown that the 𝒯-class with ρT a rectangular band congruence consists exactly of completely simple congruences. We go 1 step further and find that the 𝒯𝒱-min network and the 𝒯𝒱-network of the universal relation ω are finite. A corresponding analysis for the equality relation ϵ and a similar study for the least semilattice congruence η are also performed. Finally, we investigate the sublattice L ρ of 𝒞(S) generated by the congruences ρw where w ∈ {V, v, T, t}* and w has no subword of the form VT, TV, vt, and tv. The least lattice L whose homomorphic image is L ρ for all ρ ∈ 𝒞(S) is found explicitly and represented as a distributive lattice in terms of generators and relations.