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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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/950 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| id |
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admjournalluguniveduua-article-9502018-05-13T06:43:21Z Multi-algebras from the viewpoint of algebraic logic Cırulis, Janis cylindric algebra, linear term, multi-algebra, resolvent, singular inclusion 08A99; 03G15, 08A62 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. Lugansk National Taras Shevchenko University 2018-05-13 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/950 Algebra and Discrete Mathematics; Vol 2, No 1 (2003) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/950/479 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-05-13T06:43:21Z |
| collection |
OJS |
| language |
English |
| topic |
cylindric algebra linear term multi-algebra resolvent singular inclusion 08A99; 03G15 08A62 |
| spellingShingle |
cylindric algebra linear term multi-algebra resolvent singular inclusion 08A99; 03G15 08A62 Cırulis, Janis Multi-algebras from the viewpoint of algebraic logic |
| topic_facet |
cylindric algebra linear term multi-algebra resolvent singular inclusion 08A99; 03G15 08A62 |
| format |
Article |
| author |
Cırulis, Janis |
| author_facet |
Cırulis, Janis |
| author_sort |
Cırulis, Janis |
| title |
Multi-algebras from the viewpoint of algebraic logic |
| title_short |
Multi-algebras from the viewpoint of algebraic logic |
| title_full |
Multi-algebras from the viewpoint of algebraic logic |
| title_fullStr |
Multi-algebras from the viewpoint of algebraic logic |
| title_full_unstemmed |
Multi-algebras from the viewpoint of algebraic logic |
| title_sort |
multi-algebras from the viewpoint of algebraic logic |
| description |
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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/950 |
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AT cırulisjanis multialgebrasfromtheviewpointofalgebraiclogic |
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2025-12-02T15:37:47Z |
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2025-12-02T15:37:47Z |
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