  In this paper we introduce and study a new topologo-algebraic structure
called a (di)topological unosemigroup. This is a topological semigroup endowed
with continuous unary operations of left and right units (which have certain
continuous division property called the dicontinuity). We show that the class
of ditopological unosemigroups contains all topological groups, all topological
semilattices, all uniformizable topological unoid-semigroups, all compact
topological unosemigroups, and is closed under the operations of taking
subunosemigroup, Tychonoff product, reduced product, semidirect product, and
the Hartman-Mycielski extension.
