Embedding of graph inverse semigroups into CLP-compact topological semigroups

Ravsky²

2020

Appl. Gen. Topol.

Self Cite

<p>We investigate closed subsets (subsemigroups, resp.) of compact-like topological spaces (semigroups, resp.). We show that each Hausdorff topological space is a closed subspace of some Hausdorff ω-bounded pracompact topological space and describe open dense subspaces of<br />countably pracompact topological spaces. We construct a pseudocompact topological semigroup which contains the bicyclic monoid as a closed subsemigroup. This example provides an affirmative answer to a question posed by Banakh, Dimitrova, and Gutik in [4]. Also, we show that the semigroup of ω×ω-matrix units cannot be embedded into a Hausdorff topological semigroup whose space is weakly H-closed.</p>

Section: Applications For Topological Semigroupsmentioning

confidence: 99%

“…Namely, in [6] it was proved that for each cardinal λ > 1 a polycyclic monoid P λ does not embed as a dense subsemigroup into a feebly compact topological semigroup. In [5] embeddings of graph inverse semigroups into CLP-compact topological semigroups were described.…”

Section: Applications For Topological Semigroupsmentioning

confidence: 99%

Closed subsets of compact-like topological spaces

Ravsky²

2020

Appl. Gen. Topol.

Self Cite

“…Algebraic theory of graph inverse semigroups is well developed (see [2,5,16,17,20,22]). Topological properties of graph inverse semigroups were investigated in [6,7,8,23]. This paper is inspired by the paper of Mesyan and Mitchell [22] and can be regarded as an alternative look at the structure of graph inverse semigroups.…”

Section: Preliminariesmentioning

confidence: 99%

An alternative look at the structure of graph inverse semigroups

2019

For any graph inverse semigroup G(E) we describe subsemigroups D 0 = D ∪ {0} and J 0 = J ∪ {0} of G(E) where D and J are arbitrary D-class and J -class of G(E), respectively. In particular, we prove that for each D-class D of a graph inverse semigroup over an acyclic graph the semigroup D 0 is isomorphic to a semigroup of matrix units. Also we show that for any elements a, b of a graph inverse semigroup G(E), J a · J b ∪ J b · J a ⊂ J 0 b if there exists a path w such that s(w) ∈ J a and r(w) ∈ J b .

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

Гутік

2020

ADM

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 ω 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.