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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.
MSC: 08A99;
03G15,
08A62 Received: 09.10.2002
Language: English
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