Zero Cancellation and Equation Structure in Kiselman's Semigroup

TL;DR

This paper studies equations in Kiselman's semigroup, revealing unique solutions involving the zero element and analyzing the algebraic structure and size parity of the semigroup.

Contribution

It characterizes solutions to specific equations in Kiselman's semigroup and explores the algebraic and size properties of the semigroup.

Findings

Equation y in subsemigroup implies x y = zero only if x = zero

Equation x a_1 = zero has a solution set of size 1 + |K_{n-1}|

|K_{2n+1}| is even, |K_{2n}| is odd

Abstract

We investigate equations in Kiselman's semigroup $K_n$, generated by $a_1, \dots, a_n$. Let $f$ denote the zero element of $K_n$. We prove that if $y \in K_n$ lies in the subsemigroup generated by $a_2, \dots, a_n$, then $x y = f$ implies $x = f$. In contrast, the equation $x a_1 = f$ admits non-trivial solutions. We describe the solution set of this equation, show that its cardinality is $1 + |K_{n-1}|$, and study its algebraic structure. Moreover, we show that $|K_{2n+1}|$ is even, whereas $|K_{2n}|$ is odd.