Multi-algebras from the viewpoint of algebraic logic
Where \(U\) is a structure for a first-order language \(\mathcal{L}^\approx\) with equality \(\approx\), a standard construction associates with every formula \(f\) of \(\mathcal{L}^\approx\) the set \({\parallel}{f}{\parallel}\) of those assignments which fulfill \(f\) in \(U\). These sets ma...
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| Datum: | 2018 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/950 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | Where \(U\) is a structure for a first-order language \(\mathcal{L}^\approx\) with equality \(\approx\), a standard construction associates with every formula \(f\) of \(\mathcal{L}^\approx\) the set \({\parallel}{f}{\parallel}\) of those assignments which fulfill \(f\) in \(U\). These sets make up a (cylindric like) set algebra \(C\!s\)(\(U\)) that is a homomorphic image of the algebra of formulas. If \(\mathcal{L}^\approx\) does not have predicate symbols distinct from \(\approx\), i.e. \(U\) is an ordinary algebra, then \(C\!s \) (\(U\)) is generated by its elements \({\parallel}{s \approx t} {\parallel}\); thus, the function \((s,t) \mapsto \parallel{s \approx t}\parallel\) comprises all information on \(C\!s\)(\(U\)).In the paper, we consider the analogues of such functions for multi-algebras. Instead of \(\approx\), the relation \(\varepsilon\) of singular inclusion is accepted as the basic one (\(s \varepsilon t\) is read as `\(s\) has a single value, which is also a value of \(t\)'). Then every multi-algebra \(U\) can be completely restored from the function \((s,t) \mapsto \parallel{s \ \varepsilon \ t}\parallel\). The class of such functions is given an axiomatic description. |
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