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 $x y = f$ 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.