Year: 2025
Type: article
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 $$a^+$$ a + of all its complements is investigated as an operator on the given lattice. We can extend the definition of $$a^+$$ a + in a natural way from elements to arbitrary subsets. In particular we study the set $$a^+$$ a + for complemented modular lattices, and we characterize when the set $$a^{++}$$ a + + is a singleton. By means of the operator $$^+$$ + we introduce two other operators $$\rightarrow $$ → 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.
Source: Soft Computing
Authors: Ivan Chajda, Helmut Länger
Institutions: Palacký University Olomouc, TU Wien
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4410326612