From algebra to logic: there and back again -- the story of a hierarchy

TL;DR

This paper surveys the development of a hierarchy of logical languages linked to the quantifier alternation hierarchy in the two-variable fragment of first-order logic over linear orders.

Contribution

It provides an extensive overview of the results connecting algebraic and logical hierarchies in this specific logical framework.

Findings

Clarifies the relationship between algebraic hierarchies and logical quantifier structures

Summarizes key results in the area of two-variable logic over linear orders

Highlights open problems and future research directions

Abstract

This is an extended survey of the results concerning a hierarchy of languages that is tightly connected with the quantifier alternation hierarchy within the two-variable fragment of first order logic of the linear order.