On Brandt $\lambda^0$-extensions of semigroups with zero

Анотація

We introduce the Brandt λ⁰-extension B_λ⁰(S) of a semigroup S with zero and establish some algebraic properties of the semigroup B_λ⁰(S) with respect to the semigroup S. Also we introduce the topological Brandt λ⁰-extension of a topological semigroup S with zero and study its topological properties with respect to the topological semigroup S. In particular we show that any topological Brandt λ⁰-extension of an (absolutely) H-closed topological inverse semigroup S is (absolutely) H-closed in the class of topological inverse semigroups. Also we construct topologies on B_λ⁰(S) which preserve the absolute H-closedness and H-closedness. Using the construction of topological Brandt λ⁰-extensions of topological semigroups we give an example of absolutely H-closed metrizable inverse topological semigroup S with an absolutely H-closed ideal I such that S/I is not a topological semigroup.