Semi-linear Varieties of Lattice-Ordered Algebras

Abstract

We consider varieties of pointed lattice-ordered algebras satisfying a restricted distributivity condition and admitting a very weak implication. Examples of these varieties are ubiquitous in algebraic logic: integral or distributive residuated lattices; their \(\left\{ \cdot \right\} \)-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising from quantum logic. Given any such variety \(\fancyscript{V}\), we provide an explicit equational basis (relative to \(\fancyscript{V}\)) for the semi-linear subvariety \(\fancyscript{W}\) of \(\fancyscript{V}\). In particular, we show that if \(\fancyscript{V}\) is finitely based, then so is \(\fancyscript{W}\). To attain this goal, we make extensive use of tools from the theory of quasi-subtractive varieties.