Multi-algebras from the viewpoint of algebraic logic

Abstract

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.