Multi-algebras from the viewpoint of algebraic logic

Abstract

Where U is a structure for a first-order language L ≈ with equality ≈, a standard construction associates with every formula f of L ≈ the set kfk of those assignments which fulfill f in U. These sets make up a (cylindric like) set algebra Cs(U) that is a homomorphic image of the algebra of formulas. If L ≈ does not have predicate symbols distinct from ≈, i.e. U is an ordinary algebra, then Cs(U) is generated by its elements ks ≈ tk; thus, the function (s, t) 7→ ks ≈ tk comprises all information on Cs(U). In the paper, we consider the analogues of such functions for multi-algebras. Instead of ≈, the relation ε of singular inclusion is accepted as the basic one (sεt 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) 7→ ks ε tk. The class of such functions is given an axiomatic description.