×

Topological semigroups of matrix units. (English) Zbl 1092.22002

Summary: 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.

MSC:

22A15 Structure of topological semigroups
20M18 Inverse semigroups
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54C25 Embedding
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H10 Topological representations of algebraic systems
PDFBibTeX XMLCite