Zero Cancellation and Equation Structure in Kiselman's Semigroup

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 $xy=f$ 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 $1+\mathrm{\mid }{K}_{n-1}\mathrm{\mid }$, and study its algebraic structure. Moreover, we show that $\mathrm{\mid }{K}_{2n+1}\mathrm{\mid }$ is even, whereas $\mathrm{\mid }{K}_{2n}\mathrm{\mid }$ is odd.