Varieties of semiassociative relation algebras and tense algebras

TL;DR

This paper investigates the structure of subvariety lattices in relation algebras and semiassociative relation algebras, revealing that a particular atom in these lattices has continuum many covers, contrasting with known finite cases.

Contribution

It provides a complete classification of the covers of a specific atom in the subvariety lattice of semiassociative relation algebras, showing it has continuum many covers, unlike the finite number in related cases.

Findings

The atom in the subvariety lattice has continuum many covers.

A term equivalence between tense algebras and semiassociative r-algebras is used.

Contrasts with the finite number of covers in other atoms.

Abstract

It is well known that the subvariety lattice of the variety of relation algebras has exactly three atoms. The (join-irreducible) covers of two of these atoms are known, but a complete classification of the (join-irreducible) covers of the remaining atom has not yet been found. These statements are also true of a related subvariety lattice, namely the subvariety lattice of the variety of semiassociative relation algebras. The present article shows that this atom has continuum many covers in this subvariety lattice (and in some related subvariety lattices) using a previously established term equivalence between a variety of tense algebras and a variety of semiassociative r-algebras.