# Deciding the existence of minority terms

## TL;DR

This paper explores the complexity of determining whether a finite idempotent algebra admits a specific minority operation, revealing that the problem is NP-complete and not solvable by common polynomial-time methods.

## Contribution

It establishes the NP-completeness of deciding the existence of a minority term in finite idempotent algebras, highlighting limitations of existing polynomial-time approaches.

## Key findings

- The problem is NP-complete.
- Common polynomial-time methods are insufficient.
- Deciding minority terms is computationally hard.

## Abstract

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x) \approx m(x,y,x) \approx m(x,x,y) \approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.