Multi-algebras from the viewpoint of algebraic logic

Jānis Cīrulis

Algebra and discrete mathematics, no. 1 (2003), pp. 20-31.

Résumé

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.

Détail

Keywords: cylindric algebra, linear term, multi-algebra, resolvent, singular inclusion.