Semi-linear Varieties of Lattice-Ordered Algebras

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 {·}-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising from quantum logic. Given any such variety V, we provide an explicit equational basis (relative to V) for the semilinear subvariety W of V. In particular, we show that if V is finitely based, then so is W. To attain this goal, we make extensive use of tools from the theory of quasi-subtractive varieties.