Random equations over free semilattices

In the paper, we study equations in one variable over free semilattices. We show that the average number of solutions of a random equation over a free semilattice of a rank $n$ is equal to $\frac{3^n+2\cdot2^n}{3\cdot2^n}$. It is proved that the average number of irreducible components of algebraic sets defined by equations over a free semilattice of a countable rank is equal to 1.