On the lattice of sub-pseudovarieties of DA

TL;DR

This paper explores the structure of the lattice of sub-pseudovarieties of DA, a key algebraic structure in language theory, by leveraging known results on bands and describing a hierarchy of decidable subclasses.

Contribution

It introduces a hierarchical framework for decidable sub-pseudovarieties of DA using iterated Mal'cev products, extending understanding of this algebraic structure.

Findings

Hierarchy of decidable sub-pseudovarieties of DA established

Use of iterated Mal'cev products with definite semigroups

Enhanced understanding of DA's lattice structure

Abstract

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal'cev products with the pseudovarieties of definite and reverse definite semigroups.