Operators on complemented lattices

TL;DR

This paper explores operators on complemented lattices without explicit complementation, introducing new operators and logical connectives that generalize implication and conjunction, with applications to propositional calculus and lattice theory.

Contribution

It introduces the operator $^+$ on complemented lattices, extends it to subsets, and defines new logical operators $ o$ and $igodot$ with properties similar to intuitionistic and quantum logic.

Findings

The operator $a^{++}$ is a singleton in certain complemented modular lattices.

Operators $ o$ and $igodot$ form an adjoint pair, resembling logical implication and conjunction.

The framework relates to lattice filters and deductive systems.

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.