Brandt Extensions and Primitive Topological Inverse Semigroups

Ravsky

2013

We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup S is a closed subset of S and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt λ 0 -extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly compact topological semigroup is continuous. Also an analogue of Comfort-Ross Theorem is proved for such semigroups: a Tychonoff product of an arbitrary family of primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups is feebly compact. We describe the structure of the Stone-Čech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup S such that every maximal subgroup of S is a topological group.

Section: If X = 0 Then There Exist An Open Neighbourhood W ⊂ G Of Thesupporting

confidence: 81%

“…In the paper [7] the structure of compact and countably compact primitive topological inverse semigroups was described and was shown that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.…”

Section: Introductionmentioning

confidence: 99%

On feebly compact inverse primitive (semi)topological semigroups

Ravsky

2013

“…The constructed example gives a negative answer to Question 17 from [40]. Hclosed and absolutely H-closed (semi)topological semigroups and their extensions in different classes of topological and semitopological semigroups were studied in [8,18,19,[21][22][23][24][25][26] In [6] we showed that the λ-polycyclic monoid for an infinite cardinal λ 2 has similar algebraic properties to that of the polycyclic monoid P n with finitely many n 2 generators. In particular we proved that for every infinite cardinal λ the polycyclic monoid P λ is congruencefree, combinatorial, 0-bisimple, 0-E-unitary, inverse semigroup.…”

Section: ])mentioning

confidence: 99%

On a complete topological inverse polycyclic monoid

Bardyla

2016

Carpathian Math. Publ.

We give sufficient conditions when a topological inverse λ-polycyclic monoid P λ is absolutely Hclosed in the class of topological inverse semigroups. For every infinite cardinal λ we construct the coarsest semigroup inverse topology τ mi on P λ and give an example of a topological inverse monoid S which contains the polycyclic monoid P 2 as a dense discrete subsemigroup.Key words and phrases: inverse semigroup, bicyclic monoid, polycyclic monoid, free monoid, semigroup of matrix units, topological semigroup, topological inverse semigroup, minimal topology.Ivan Franko National University, 1 Universytetska str., 79000, Lviv, Ukraine E-mail: sbardyla@yahoo.com (Bardyla S.O.), o_gutik@franko.lviv.ua, ovgutik@yahoo.com (Gutik O.V.) In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [10,12,16,31]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By N we denote the set of all positive integers and by ω the first infinite cardinal.A semigroup S is called an inverse semigroup if every a in S possesses a unique inverse, i.e. if there exists a unique element a −1 in S such thatA map that associates to any element of an inverse semigroup its inverse is called the inversion.A band is a semigroup of idempotents. If S is a semigroup, then we shall denote the subset of idempotents in S by E(S). If S is an inverse semigroup, then E(S) is closed under multiplication. The semigroup operation on S determines the following partial order on E(S): e f if and only if e f = f e = e. This order is called the natural partial order on E(S). A semilattice is a commutative semigroup of idempotents. A semilattice E is called linearly ordered or a chain if its natural order is a linear order. A maximal chain of a semilattice E is a chain which is properly contained in no other chain of E. The Axiom of Choice implies the existence of maximal chains in any partially ordered set. According to [35, Definition II.5.12] a chain L is called ω-chain if L is order isomorphic to {0, −1, −2, −3, . . .} with the usual order . Let E be a semilattice and e ∈ E. We denote ↓e = { f ∈ E | f e} andIf S is a semigroup, then we shall denote by R, L , D and H the Green relations on S (see [17] The R-class (resp., L -, H -, or D-class) of the semigroup S which contains an element a of S will be denoted by R a (resp., L a , H a , or D a ).

“…. Then by Proposition 12 from [2], U(0) is an open-and-closed subset of S. Thus we have that the map f : S → [0, 1] defined by the formula…”

Section: Primitive Pseudocompact Topological Inverse Semigroupsmentioning

confidence: 94%

Pseudocompact Primitive Topological Inverse Semigroups

Pavlyk

2014

J Math Sci

In the paper we study pseudocompact primitive topological inverse semigroups. We describe the structure of pseudocompact primitive topological inverse semigroups and show that the Tychonoff product of a family of pseudocompact primitive topological inverse semigroups is a pseudocompact topological space. Also we prove that the Stone-Čech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup.