On a complete topological inverse polycyclic monoid

Abstract

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 ).

Embedding of graph inverse semigroups into CLP-compact topological semigroups

“…Similar arguments imply that the semigroup T is isomorphic to the semigroup of N×N-matrix units and T is countably pracompact. This contradicts [13,Theorem 4.4]. Hence the set C is empty as well.…”

“…Put . Then T is countably compact at the dense discrete subspace T \ {0} which contradicts [13,Theorem 4.4] where it was proved that infinite semigroup of matrix units can not be embedded densely into a feebly compact topological semigroup (recall that countably pracompact topological spaces are feebly compact). Hence the set B is empty.…”

“…Algebraic theory of graph inverse semigroups is well developed (see [2,8,10,26,27,30,32]). Topological properties of graph inverse semigroups were investigated in [9,11,12,13,33].…”

An alternative look at the structure of graph inverse semigroups

“…Polycyclic monoid P k over a finite non-zero cardinal k was introduced in [24]. Algebraic and topological properties of polycyclic monoids were investigated in [4,9,10,13,20,21].…”

On locally compact semitopological graph inverse semigroups

“…Algebraic properties of a semigroup P k were investigated in [20] and [21]. Algebraic and topological properties of the λ-polycyclic monoid were investigated in [8] and [9]. In particular, it was proved that for every non-zero cardinal λ the only locally compact semigroup topology on the λ-polycyclic monoid is the discrete topology.…”