Operators on complemented lattices

Abstract

The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping $^+$ assigning to each element $a$ the set $a^+$ of all its complements is investigated as an operator on the given lattice. We can extend the definition of $a^+$ in a natural way from elements to arbitrary subsets. In particular we study the set $a^+$ for complemented modular lattices, and we characterize when the set $a^{++}$ is a singleton. By means of the operator $^+$ we introduce two other operators $\to$ and $\odot$ which can be considered as implication and conjunction in a certain propositional calculus, respectively. These two logical connectives are ``unsharp'' which means that they assign to each pair of elements a non-empty subset. However, also these two derived operators share a lot of properties with the corresponding logical connectives in intuitionistic logic or in the logic of quantum mechanics. In particular, they form an adjoint pair. Finally, we define so-called deductive systems and we show their relationship to the mentioned operators as well as to lattice filters.