Topological semigroups of matrix units

We prove that the semigroup of matrix units is stable. Compact, countably compact and pseudocompact topologies $\tau$ on the infinite semigroup of matrix units $B_\lambda$ such that $(B_\lambda,\tau)$ is a semitopological (inverse) semigroup are described. We prove the following properties of an infinite topological semigroup of matrix units. On the infinite semigroup of matrix units there exists no semigroup pseudocompact topology. Any continuous homomorphism from the infinite topological semigroup of matrix units into a compact topological semigroup is annihilating. The semigroup of matrix units is algebraically $h$-closed in the class of topological inverse semigroups. Some $H$-closed minimal semigroup topologies on the infinite semigroup of matrix units are considered.