Pointlike sets for varieties determined by groups

Abstract

For a variety of finite groups H, let H denote the variety of finite semigroups all of whose subgroups lie in H. We give a characterization of the subsets of a finite semigroup that are pointlike with respect to H. Our characterization is effective whenever H has a decidable membership problem. In particular, the separation problem for H-languages is decidable for any decidable variety of finite groups H. This generalizes Henckell's theorem on decidability of aperiodic pointlikes.Pointlike sets. Let V be a variety of finite semigroups. A subset X of a finite semigroup S is called V-pointlike if, for any relational morphism ϕ : S → T with T ∈ V, there exists t ∈ T such that X ⊆ tϕ −1 . Any singleton set is V-pointlike, and the collection of V-pointlike subsets of a semigroup S forms a downward closed subsemigroup PL V (S) of the power semigroup 2 S . Moreover, the assignment S → PL V (S) has the following 'monadic' property first observed by Henckell and Rhodes.Proposition 2.1. Let S be a finite semigroup. For any V-pointlike subset X of the semigroup PL V (S), the union X is a V-pointlike subset of S.Proof. Suppose that X is V-pointlike in PL V (S), and let ϕ : S → V be a relational morphism with V ∈ V. We show that there exists v ∈ V such that X ⊆ vϕ −1 . Define the relation Φ : PL V (S) → V by setting XΦ := {v ∈ V | X ⊆ vϕ −1 } for

First-order separation over countable ordinals

“…The computability of pointlike sets was shown to be equivalent to the decidability of the covering problem by Almeida [Alm99]. Alternative proofs of separation and covering problems for FO were given recently in [PZ16,vGS19], and, ever since Henckell's work, the computability of FO-pointlike sets was also extended to pointlike sets for other varieties-for example [Ash91] for the variety of finite groups, [AZ97] for the variety of J-trivial finite semigroups and [GS19] for varieties of finite semigroups determined by a variety of finite groups; also see [GS19] for further references. Place and Zeitoun recently used pointlike sets, in the form of covering problems [PZ18b], to resolve long-standing open membership problems for the lower levels of the dot-depth and of the Straubing-Thérien hierarchies [PZ18a,PZ19,PZ21].…”