A General Theory of Pointlike Sets

TL;DR

This paper develops a comprehensive theoretical framework for understanding pointlike sets using lattice theory and Galois connections, offering new characterizations and methods for analyzing their properties without relational morphisms.

Contribution

It introduces a unifying lattice-based framework for pointlike sets, characterizes pointlike functors as fixed points of a closure operator, and provides a method to transfer lower bounds across pseudovarieties.

Findings

Pointlike functors are fixed points of a closure operator.

A characterization of pointlikes independent of relational morphisms.

A method for transferring lower bounds along continuous operators.

Abstract

We introduce a general unifying framework for the investigation of pointlike sets. The pointlike functors are considered as distinguished elements of a certain lattice of subfunctors of the power semigroup functor; in particular, we exhibit the pointlike functors as the fixed points of a closure operator induced by an antitone Galois connection between this lattice of functors and the lattice of pseudovarieties. Notably, this provides a characterization of pointlikes which does not mention relational morphisms. Along the way, we formalize various common heuristics and themes in the study of pointlike sets. As an application, we provide a general method for transferring lower bounds for pointlikes along a large class of continuous operators on the lattice of pseudovarieties.