  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.
