Terms that define nuclei on residuated lattices: a case study of BL-algebras
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910764963463168 |
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| author | Buss, Sebastián Castaño, Diego Varela, José Patricio Díaz |
| author_facet | Buss, Sebastián Castaño, Diego Varela, José Patricio Díaz |
| contents | A nucleus $γ$ on a (bounded commutative integral) residuated lattice $\mathbf{A}$ is a closure operator that satisfies the inequality $γ(a) \cdot γ(b) \leq γ(a \cdot b)$ for all $a,b \in A$. In this article, among several results, a description of an arbitrary nucleus on a residuated lattice is given. Special attention is given to terms that define a nucleus on every structure of a variety, as a means of generalizing the double negation operation. Some general results about these terms are presented, together with examples. The main result of this article consists of the description of all terms of this kind for every given subvariety of BL-algebras. We exhibit interesting nontrivial examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Terms that define nuclei on residuated lattices: a case study of BL-algebras Buss, Sebastián Castaño, Diego Varela, José Patricio Díaz Logic A nucleus $γ$ on a (bounded commutative integral) residuated lattice $\mathbf{A}$ is a closure operator that satisfies the inequality $γ(a) \cdot γ(b) \leq γ(a \cdot b)$ for all $a,b \in A$. In this article, among several results, a description of an arbitrary nucleus on a residuated lattice is given. Special attention is given to terms that define a nucleus on every structure of a variety, as a means of generalizing the double negation operation. Some general results about these terms are presented, together with examples. The main result of this article consists of the description of all terms of this kind for every given subvariety of BL-algebras. We exhibit interesting nontrivial examples. |
| title | Terms that define nuclei on residuated lattices: a case study of BL-algebras |
| topic | Logic |
| url | https://arxiv.org/abs/2412.19690 |