Bibliographic Details
| Title: |
Operators on complemented lattices. |
| Authors: |
Chajda, Ivan1 (AUTHOR) ivan.chajda@upol.cz, Länger, Helmut1,2 (AUTHOR) helmut.laenger@tuwien.ac.at |
| Source: |
Soft Computing - A Fusion of Foundations, Methodologies & Applications. Apr2025, Vol. 29 Issue 7, p3115-3123. 9p. |
| Subjects: |
Propositional calculus, Quantum mechanics, Definitions |
| 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 → and ⊙ 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. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
