Algebras and varieties where Sasaki operations form an adjoint pair

Abstract

The so-called Sasaki projection was introduced by U. Sasaki on the lattice L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H) onto a certain sublattice of L(H). Since L(H) is an orthomodular lattice, the Sasaki projection and its dual can serve as the logical connectives conjunction and implication within the logic of quantum mechanics. It was shown by the authors in a previous paper that these operations form a so-called adjoint pair. The natural question arises if this result can be extended also to lattices with a unary operation which need not be orthomodular or to other algebras with two binary and one unary operation. To show that this is possible is the aim of the present paper. We determine a variety of lattices with a unary operation where the Sasaki operations form an adjoint pair and we continue with so-called $\lambda$-lattices and certain classes of semirings. We show that the Sasaki operations have a deeper sense than originally assumed by their author and can be applied also outside the lattices of closed linear subspaces of a Hilbert space.