On closures in semitopological inverse semigroups with continuous inversion

We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group G is H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if G is compact, a Hausdorff linearly ordered topological semilattice E is H-closed in the class of semitopological semilattices if and only if E is H-closed in the class of topological semilattices, and a topological Brandt λ⁰-extension of S is (absolutely) H-closed in the class of semitopological inverse semigroups with continuous inversion if and only if so is S. Also, we construct an example of an H-closed non-absolutely H-closed semitopological semilattice in the class of semitopological semilattices

Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.Comment: 4 page

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology τ on the bicyclic monoid with adjoined zero C⁰ is called weak if it is contained in the coarsest inverse semigroup topology on C⁰. We show that the lattice W of all weak shift-continuous topologies on C⁰ is isomorphic to the lattice SIF¹×SIF¹ where SIF¹ is the set of all shift-invariant filters on ! with an attached element 1 endowed with the following partial order: F ≤ G if and only if G = 1 or F ⊂ G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2ᶜ and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t

On a semitopological polycyclic monoid

We study algebraic structure of the λ-polycyclic monoid Pλ and its topologizations. We show that the λ-polycyclic monoid for an infinite cardinal λ≥2 has similar algebraic properties so has the polycyclic monoid Pn with finitely many n≥2 generators. In particular we prove that for every infinite cardinal λ the polycyclic monoid Pλ is a congruence-free combinatorial 0-bisimple 0-E-unitary inverse semigroup. Also we show that every non-zero element x is an isolated point in (Pλ,τ) for every Hausdorff topology τ on Pλ, such that (Pλ,τ) is a semitopological semigroup, and every locally compact Hausdorff semigroup topology on Pλ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies τ on Pλ such that (Pλ,τ) is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal λ≥2 any continuous homomorphism from a topological semigroup Pλ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains Pλ as a dense subsemigroup

On monoids of monotone injective partial selfmaps of Ln ×lex Z with co-finite domains and images

We study the semigroup IO∞(Zⁿlex) of monotone injective partial selfmaps of the set of Ln × lex Z having co-finite domain and image, where Ln ×lex Z is the lexicographic product of n-elements chain and the set of integers with the usual order. We show that IO∞(Zⁿlex) is bisimple and establish its projective congruences. We prove that IO∞(Zⁿlex) is finitely generated, and for n = 1 every automorphism of IO∞(Zⁿlex) is inner and show that in the case n ⩾ 2 the semigroup IO∞(Zⁿlex) has non-inner automorphisms. Also we show that every Baire topology τ on IO∞(Znlex) such that (IO∞(Znlex),τ) is a Hausdorff semitopological semigroup is discrete, construct a non-discrete Hausdorff semigroup inverse topology on IO∞(Zⁿlex), and prove that the discrete semigroup IO∞(Zⁿlex) cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup S is an ideal in S

On H-closed topological semigroups and semilattices

In this paper, we show that if S is an H-closed topological semigroup and e is an idempotent of S, then eSe is an H-closed topological semigroup. We give sufficient conditions on a linearly ordered topological semilattice to be H-closed. Also we prove that any H-closed locally compact topological semilattice and any H-closed topological weakly U-semilattice contain minimal idempotents. An example of a countably compact topological semilattice whose topological space is H-closed is constructed

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\operatorname{inv}\colon H(S)\to H(S)$ is continuous; and (iv) the projection $\pi\colon H(S)\to E(S)$, $\pi\colon x\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous

On chains in HH-closed topological pospaces

We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.Comment: We have rewritten and substantially expanded the manuscrip