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

O. V. Gutik, K. P. Pavlyk


We introduce the Brandt λ0-extension Bλ0(S) of a semigroup S with zero and establish some algebraic properties of the semigroup Bλ0(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 Bλ0(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.

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