The variety of complemented lattices where the Sasaki operations form an adjoint pair

TL;DR

This paper characterizes complemented lattices where Sasaki operations form an adjoint pair, proving they form a variety with specific algebraic properties and providing a finite basis for related ideal terms.

Contribution

It establishes that complemented lattices with Sasaki operations forming an adjoint pair constitute a variety, and details their algebraic structure and congruence properties.

Findings

The class of such lattices forms a variety.

This variety is congruence permutable and regular.

A finite basis for ideal terms is provided.

Abstract

The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sasaki operations were introduced by the second two authors. In a previous paper there are described several classes of lattices, $\lambda$-lattices and semirings where the Sasaki operations form an adjoint pair. In the present paper we prove that the class of complemented lattices with this property forms a variety and we explicitly state its defining identities. Moreover, we prove that this variety V is congruence permutable and regular. Hence every ideal I of some member L of V is a kernel of some congruence on L. Finally, we determine a finite basis of so-called ideal terms and describe the congruence $\Theta_I$ determined by the ideal I.