A General Theory of Pointlike Sets
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.
Cite
@article{arxiv.2108.12824,
title = {A General Theory of Pointlike Sets},
author = {Karsten Henckell and Samuel Herman},
journal= {arXiv preprint arXiv:2108.12824},
year = {2021}
}
Comments
54 pages