Polynomial-time right-ideal morphisms and congruences

TL;DR

This paper explores algebraic structures related to the P versus NP problem, constructing specific monoids and their homomorphic images, revealing connections to well-known groups like Thompson's V, and analyzing their regularity properties.

Contribution

It introduces a new finitely generated, J^0-simple homomorphic image of RM^P, linking algebraic properties to the P versus NP question and connecting to Thompson's group V.

Findings

M^P_poly is finitely generated and J^0-simple.

M^P_poly is non-regular iff NP is not P.

The group of units of M^P_poly is Thompson's V.

Abstract

We continue with the functional approach to the P-versus-NP problem, begun in [2, 3]. We previously constructed a monoid RM^P that is non-regular iff NP is not P. We now construct homomorphic images of RM^P with interesting properties. In particular, the homomorphic image M^P_poly of RM^P is finitely generated and J^0-simple, and is non-regular iff NP is not P. The group of units of M^P_poly is the famous Richard Thompson group V.