Multi-algebras from the viewpoint of algebraic logic

Jānis Cīrulis

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

Résumé

Where $\mathit{U}$ 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 $\Vert f\Vert$ of those assignments which fulfill $f$ in $\mathit{U}$. These sets make up a (cylindric like) set algebra $Cs(\mathit{U})$ that is a homomorphic image of the algebra of formulas. If ${\mathcal{L}}^{\approx }$ does not have predicate symbols distinct from $\approx$, i.e. $\mathit{U}$ is an ordinary algebra, then $Cs(\mathit{U})$ is generated by its elements $\Vert s\approx t\Vert$; thus, the function $(s,t)\mapsto \Vert s\approx t\Vert$ comprises all information on $Cs(\mathit{U})$. In the paper, we consider the analogues of such functions for multi-algebras. Instead of $\approx$, the relation $\epsilon$ of singular inclusion is accepted as the basic one ($s\epsilon t$ is read as `$s$ has a single value, which is also a value of $t$'). Then every multi-algebra $\mathit{U}$ can be completely restored from the function $(s,t)\mapsto \Vert s\epsilon t\Vert$. The class of such functions is given an axiomatic description.

Détail

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