Varieties and quasivarieties of lattices with complementation

We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses satisfying additional conditions, such as the quasi-identity $(x^{\mathrm{\prime }}\wedge y\approx 0\mathrm{\&}x\wedge y^{\mathrm{\prime }}\approx 0)$ $\Rightarrow x\approx y$, modularity, or De Morgan’s laws. We present a construction resembling a semidirect product that yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying this quasi-identity. We axiomatize small varieties, each of which covers the variety of Boolean algebras, generated by certain small modular lattices with De Morgan complementation.