On equations over Brandt semigroups

TL;DR

This paper investigates equations over Brandt semigroups, revealing asymptotic behaviors of unsolvable equations and average solutions, contributing to the algebraic understanding of these structures.

Contribution

It provides the first asymptotic analysis of the solvability and solutions of equations over Brandt semigroups.

Findings

Number of unsolvable equations in one variable asymptotically equals 2/n^2

Average number of solutions asymptotically equals n^2

Results enhance understanding of algebraic equations in semigroup theory

Abstract

In this paper, we study equations over Brandt semigroup $B_n$. We compute that the number of unsolvable equations in one variable asymptotically equals $\frac{2}{n^2}$, and the average number of solutions of these equations asymptotically equals $n^2$.