Logics to which the class of neat reducts is sensitive to

TL;DR

This paper investigates how the class of neat reducts in algebraic logic is affected by different logical languages, showing sensitivity to certain infinitary and finitary logics depending on the dimension.

Contribution

It demonstrates that the class of neat reducts is sensitive to quantifier-free infinitary logics in infinite dimensions, and to finitary logics in finite dimensions, revealing logical boundaries of algebraic classes.

Findings

Neat reducts are sensitive to quantifier-free infinitary logics in infinite dimensions.

In finite dimensions, neat reducts are sensitive without requiring infinite conjunctions.

The sensitivity indicates non-elementarity of the class of neat reducts in various logical contexts.

Abstract

Let L be a quantifier predicate logic. Let K be a class of algebras. We say that K is sensitive to L, if there is an algebra in K, that is L interpretable into an another algebra, and this latter algebra is elementary equivalent to an algebra not in K. (In particular, if L is L_{\omega,\omega}, this means that K is not elementary). We show that the class of neat reducts of every dimension is sensitive to quantifier free predicate logics with infinitary conjunctions; for finite dimensions, we do not need infinite conjunctions.