Topological Inverse Semigroups Research Articles

Topological embeddings into transformation monoids

Abstract In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ℕ ℕ {\mathbb{N}^{\mathbb{N}}} or the symmetric inverse monoid I ℕ {I_{\mathbb{N}}} with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and I ℕ {I_{\mathbb{N}}} . We construct several examples of countable Polish topological semigroups that do not embed into ℕ ℕ {\mathbb{N}^{\mathbb{N}}} , which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of ℕ ℕ {\mathbb{N}^{\mathbb{N}}} . The former complements recent works of Banakh et al.

On the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, which is generated by the family $\mathscr{F}_n$ of finite bounded intervals of $\omega$

We study the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, which is introduced in the paper [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19 (in Ukrainian)], in the case when the ${\omega}$-closed family $\mathscr{F}_n$ generated by the set $\{0,1,\ldots,n\}$. We show that the Green relations $\mathscr{D}$ and $\mathscr{J}$ coincide in $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is isomorphic to the semigroup $\mathscr{I}_\omega^{n+1}(\overrightarrow{\mathrm{conv}})$ of partial convex order isomorphisms of $(\omega,\leqslant)$ of the rank $\leqslant n+1$, and $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$. In particular, we prove that for any shift-continuous $T_1$-topology $\tau$ on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ every non-zero element of $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is an isolated point of $(\boldsymbol{B}_{\omega}^{\mathscr{F}_n},\tau)$, $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every $\omega_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is $H$-closed in the class of Hausdorff topological semigroups.

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

Abstract Given a partial action π of an inverse semigroup S on a ring 𝒜 {\mathcal{A}} , one may construct its associated skew inverse semigroup ring 𝒜 ⋊ π S {\mathcal{A}\rtimes_{\pi}S} . Our main result asserts that, when 𝒜 {\mathcal{A}} is commutative, the ring 𝒜 ⋊ π S {\mathcal{A}\rtimes_{\pi}S} is simple if, and only if, 𝒜 {\mathcal{A}} is a maximal commutative subring of 𝒜 ⋊ π S {\mathcal{A}\rtimes_{\pi}S} and 𝒜 {\mathcal{A}} is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra A R ⁢ ( 𝒢 ) {A_{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid 𝒢 {\mathcal{G}} .

Automatic continuity of homomorphisms between topological inverse semigroups

Abstract We find conditions on topological inverse semigroups X, Y guaranteeing the continuity of any homomorphism h : X → Y having continuous restrictions to any subsemilattice and any subgroup of X.

On a complete topological inverse polycyclic monoid

We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. For every infinite cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid $S$ which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.

Pseudocompact Primitive Topological Inverse Semigroups

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 – Cech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup.

On topological Brandt semigroups

We describe the structure of pseudocompact completely 0 -simple topological inverse semigroups. We also give sufficient conditions under which the topological Brandt λ0 -extensions of an (absolutely) H-closed semigroup are (absolutely) H -closed semigroups.

On the closure of the extended bicyclic semigroup

In the paper we study the semigroup CZ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup CZ and prove that every non-trivial congruence C on the semigroup CZ is a group congruence, and moreover the quotient semigroup CZ=C is isomorphic to a cyclic group. Also we show that the semigroup CZ as a Hausdor semitopological semigroup admits only the discrete topology. Next we study the closure clT (CZ) of the semigroup CZ in a topological semigroup T. We show that the non-empty remainder of CZ in a topological inverse semigroup T consists of a group of units H(1T) of T and a two-sided ideal I of T in the case when H(1T) 6 ? and I 6 ?. In the case when T is a locally compact topological inverse semigroup and I 6 ? we prove that an ideal I is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup CZ [ I. Also we show that if the group of units H(1T) of the semigroup T is non-empty, then H(1T) is either singleton or H(1T) is topologically isomorphic to the discrete additive group of integers.

Topological monoids of monotone injective partial selfmaps of ℕ with cofinite domain and image

In this paper we study the semigroup ℐ ∞↗ (ℕ) of partial cofinal monotone bijective transformations of the set of positive integers ℕ. We show that the semigroup ℐ ∞↗ (ℕ) has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology τ on ℐ ∞↗ (ℕ) such that (ℐ ∞↗ (ℕ); τ) is a topological inverse semigroup, is discrete. Finally, we describe the closure of (ℐ ∞↗ (ℕ); τ) in a topological semigroup.

Symmetric inverse topological semigroups of finite rank ≤ n

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.

Brandt Extensions and Primitive Topological Inverse Semigroups

We study (countably) compact and (absolutely) H‐closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.

On the Brandt λ 0-extensions of monoids with zero

We study algebraic properties of the Brandt λ 0-extensions of monoids with zero and non-trivial homomorphisms between the Brandt λ 0-extensions of monoids with zero. We introduce finite, compact topological Brandt λ 0-extensions of topological semigroups and countably compact topological Brandt λ 0-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt λ 0-extensions of topological monoids with zero. We also describe a category whose objects are ingredients in the constructions of finite (compact, countably compact) topological Brandt λ 0-extensions of topological monoids with zeros.

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B ( −∞, ∞ ) 1 , a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}.

An algebraic generalization of Kripke structures

AbstractThe Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I n λ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.

Universal Objects in Some Classes of Clifford Topological Inverse Semigroups

We construct universal objects in various classes of Clifford topological inverse semigroups. In particular, we show that each compact Abelian topological inverse semigroup with Lawson maximal semilattice embeds into a power of the cone over the circle.

On Countably Compact 0-Simple Topological Inverse Semigroups

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.

On Topological Semigroups of Matrix Units

On the infinite semigroup of matrix units there exists no semigroup compact [countably compact] 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.

On cardinal invariants and metrizability of topological inverse semigroups

Let S be a topological inverse semigroup, E={x∈S: xx=x} be the maximal semilattice in S, and C={x∈S: xe=ex for every idempotent e∈ E} be the maximal Clifford semigroup of S. It is proven that a Lindelöf locally compact semigroup S is metrizable if and only if the maximal Clifford semigroup C is metrizable. We derive from this that a compact topological inverse semigroup S is metrizable, provided the maximal semilattice E and all maximal groups of S are metrizable and one of the following conditions is satisfied: (1) (MA+¬CH) holds; (2) E is a G δ -set in the maximal Clifford semigroup C of S; (3) E is a Lawson semilattice; (4) all maximal groups of C are Lie groups; (5) S is dyadic or scadic compact; (6) S is a fragmentable (or Rosenthal) monolithic compactum; (7) S is a Corson (or Rosenthal) compactum with countable spread.

Embedding of Countable Topological Semigroups in Simple Countable Connected Topological Semigroups

We prove that any countable Hausdorff topological (inverse) semigroup is topologically isomorphically embedded into a simple countable connected Hausdorff topological (inverse) semigroup with identity.