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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.11490 |
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| _version_ | 1866912380088221696 |
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| author | Abbadini, Marco Přenosil, Adam |
| author_facet | Abbadini, Marco Přenosil, Adam |
| contents | The theory of natural dualities provides a well-developed framework for studying Stone-like dualities induced by an algebra $\mathbf{L}$ which acts as a dualizing object when equipped with suitable topological and relational structure. The development of this theory has, however, largely remained restricted to the case where $\mathbf{L}$ is finite. Motivated by the desire to provide a universal algebraic formulation of the existing duality of Cignoli and Marra or locally weakly finite MV-algebras and to extend it to a corresponding class of positive MV-algebras, in this paper we investigate Stone-like dualities where the algebra $\mathbf{L}$ is allowed to be infinite. This requires restricting our attention from the whole prevariety generated by $\mathbf{L}$ to the subclass of algebras representable as algebras of $\mathbf{L}$-valued functions of finite range, a distinction that does not arise in the case of finite $\mathbf{L}$. Provided some requirements on $\mathbf{L}$ are met, our main result establishes a categorical duality for this class of algebras, which covers the above cases of MV-algebras and positive MV-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duality for finitely valued algebras Abbadini, Marco Přenosil, Adam Logic The theory of natural dualities provides a well-developed framework for studying Stone-like dualities induced by an algebra $\mathbf{L}$ which acts as a dualizing object when equipped with suitable topological and relational structure. The development of this theory has, however, largely remained restricted to the case where $\mathbf{L}$ is finite. Motivated by the desire to provide a universal algebraic formulation of the existing duality of Cignoli and Marra or locally weakly finite MV-algebras and to extend it to a corresponding class of positive MV-algebras, in this paper we investigate Stone-like dualities where the algebra $\mathbf{L}$ is allowed to be infinite. This requires restricting our attention from the whole prevariety generated by $\mathbf{L}$ to the subclass of algebras representable as algebras of $\mathbf{L}$-valued functions of finite range, a distinction that does not arise in the case of finite $\mathbf{L}$. Provided some requirements on $\mathbf{L}$ are met, our main result establishes a categorical duality for this class of algebras, which covers the above cases of MV-algebras and positive MV-algebras. |
| title | Duality for finitely valued algebras |
| topic | Logic |
| url | https://arxiv.org/abs/2505.11490 |