The variety of complemented lattices where conjunction and implication form an adjoint pair
Václav Cenker, Ivan Chajda, Helmut Länger · Journal of Logic and Computation · 2025
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 [4] and [6]. In [6] 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 $\mathcal V$ is congruence permutable and regular. Hence every ideal $I$ of some member $\mathbf L$ of $\mathcal V$ is a kernel of some congruence on $\mathbf L$. Finally, we determine a finite basis of so-called ideal terms and describe the congruence $\varTheta _{I}$ determined by the ideal $I$.