Operators on complemented lattices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914832270229504 |
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| author | Chajda, Ivan Länger, Helmut |
| author_facet | Chajda, Ivan Länger, Helmut |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operators on complemented lattices Chajda, Ivan Länger, Helmut Logic 06C15, 06C05, 06C20 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. |
| title | Operators on complemented lattices |
| topic | Logic 06C15, 06C05, 06C20 |
| url | https://arxiv.org/abs/2406.07665 |