  We study algebraic properties of the Brandt $\lambda^0$-extensions of monoids
with zero and non-trivial homomorphisms between the Brandt
$\lambda^0$-extensions of monoids with zero. We introduce finite, compact
topological Brandt $\lambda^0$-extensions of topological semigroups and
countably compact topological Brandt $\lambda^0$-extensions of topological
inverse semigroups in the class of topological inverse semigroups and establish
the structure of such extensions and non-trivial continuous homomorphisms
between such topological Brandt $\lambda^0$-extensions of topological monoids
with zero. We also describe a category whose objects are ingredients in the
constructions of finite (compact, countably compact) topological Brandt
$\lambda^0$-extensions of topological monoids with zeros.
