2024/06/11 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1 · 1 citation
Mathematics · #06C05 #06C15 #06C20 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2406.07665
openalex publication_date 2024/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → 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.