SMB algebras II: On the Constraint Satisfaction Problem over Semilattices of Mal'cev Blocks

TL;DR

This paper studies SMB algebras, a class of semilattices with Mal'cev blocks, demonstrating their role in inducing tractable templates for the Constraint Satisfaction Problem and comparing key proofs of the CSP Dichotomy.

Contribution

It reestablishes that all SMB algebras induce tractable CSP templates and analyzes the similarities between two major proofs of the CSP Dichotomy.

Findings

SMB algebras induce tractable CSP templates

Two proofs of the CSP Dichotomy are more similar than previously thought for SMB algebras

The paper sets the stage for further research on proof similarities in the CSP Dichotomy

Abstract

We define a class of algebras, the semilattices of Mal'cev blocks (for short, SMB algebras). In a nutshell, these algebras are semilattices in which each element gets blown up into a Mal'cev algebra. We publish for the first time our old proofs that some SMB algebras induce tractable templates of the reprove that the Constraint Satisfaction Problem. Next, we reprove that, in fact, all SMB algebras induce tractable templates of the Constraint Satisfaction Problem, a result already proved by A. Bulatov. Also, we compare the two general proofs of the CSP Dichotomy and prove they are more similar than initially thought when they are applied to SMB algebras. This paper is the second in the series of papers investigating the SMB algebras and it is a precursor to our further research on the similarities between the proofs of the Dichotomy Theorem.