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Algebra and Discrete Mathematics, 2003, Issue 1, Pages 20–31
(Mi adm366)
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This article is cited in 1 scientific paper (total in 1 paper)
RESEARCH ARTICLE
Multi-algebras from the viewpoint of algebraic logic
Jānis Cīrulis Department of Computer Science, University
of Latvia, Raiņna b., 19, LV–1586 Riga,
Latvia
Abstract:
Where $\boldsymbol U$ 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 $\| f\|$ of those assignments which fulfill $f$ in $\boldsymbol U$. These sets make up a (cylindric like) set algebra $Cs(\boldsymbol U)$ that is a homomorphic image of the algebra of formulas. If $\mathcal L^\approx$ does not have predicate symbols distinct from $\approx$, i.e. $\boldsymbol U$ is an ordinary algebra, then $Cs(\boldsymbol U)$ is generated by its elements $\| s\approx t\|$; thus, the function $(s,t) \mapsto\|s\approx t\|$ comprises all information on $Cs(\boldsymbol U)$.
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 $\boldsymbol U$ can be completely restored from the function $(s,t)\mapsto\|s\varepsilon t\|$. The class of such functions is given an axiomatic description.
Keywords:
cylindric algebra, linear term, multi-algebra, resolvent, singular inclusion.
Received: 09.10.2002
Citation:
Jānis Cīrulis, “Multi-algebras from the viewpoint of algebraic logic”, Algebra Discrete Math., 2003, no. 1, 20–31
Linking options:
https://www.mathnet.ru/eng/adm366 https://www.mathnet.ru/eng/adm/y2003/i1/p20
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