The Pseudofinite Monadic Second Order Theory of Linear Order

TL;DR

This paper investigates the pseudofinite monadic second order theory of finite linear orders, providing explicit axioms, characterizations, and connections to profinite algebra using extended Stone duality.

Contribution

It introduces a recursive axiomatization of the theory and links it to profinite algebra, advancing understanding of monadic second order logic over linear orders.

Findings

Explicit recursive axioms for the theory

Characterization of completions via residue sequences

Connection established with the free profinite monoid

Abstract

Monadic second order logic is the expansion of first order logic by quantifiers ranging over unary relations. We study the shared monadic second order theory of finite linear orders, i.e. the pseudofinite monadic second order theory of linear order, using a first order setup . We give explicit (and recursive) axioms and characterise the completions in terms of residue sequences. A connection with profinite algebra, in particular with the free profinite monoid on one generator, is established via extended Stone duality.