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This study presents a novel approach to defining topological rough semigroups on an approximation space. The concepts of topological space and rough semigroup are naturally combined to achieve this goal. Also, some basic results and examples are presented. Furthermore, some compactness properties are also studied. In addition, their rough subsemigroups and rough ideals are analysed. [ABSTRACT FROM AUTHOR]

This study presents a novel approach to defining topological rough semigroups on an approximation space. The concepts of topological space and rough semigroup are naturally combined to achieve this goal. Also, some basic results and examples are presented. Furthermore, some compactness properties are also studied. In addition, their rough subsemigroups and rough ideals are analysed.

We give a pointwise version of sensitivity in terms of open covers for a semiflow (T, X) of a topological semigroup T on a Hausdorff space X and call it a Hausdorff sensitive point. If (X , U) is a uniform space with topology τ , then the definition of Hausdorff sensitivity for (T , (X , τ)) gives a pointwise version of sensitivity in terms of uniformity and we call it a uniformly sensitive point. For a semiflow (T, X) on a compact Hausdorff space X, these notions (i.e. Hausdorff sensitive point and uniformly sensitive point) are equal and they are T-invariant if T is a C-semigroup. They are not preserved by factor maps and subsystems, but behave slightly better with respect to lifting. We give the definition of a topologically equicontinuous pair for a semiflow (T, X) on a topological space X and show that if (T, X) is a topologically equicontinuous pair in (x, y), for all y ∈ X , then Tx ¯ = D T (x) where D T (x) = ⋂ { TU ¯ : for all open neighborhoods U of x }. We prove for a topologically transitive semiflow (T, X) of a C-semigroup T on a regular space X with a topologically equicontinuous point that the set of topologically equicontinuous points coincides with the set of transitive points. This implies that every minimal semiflow of C-semigroup T on a regular space X with a topologically equicontinuous point is topologically equicontinuous. Moreover, we show that if X is a regular space and (T, X) is not a topologically equicontinuous pair in (x, y), then x is a Hausdorff sensitive point for (T, X). Hence, a minimal semiflow of a C-semigroup T on a regular space X is either topologically equicontinuous or topologically sensitive. [ABSTRACT FROM AUTHOR]

We recall various notions of size for a topological semigroup T, not necessarily discrete, such as GH-syndetic set, syndetic set, and positive Følner density set. We give new information about these sets and give some examples to study the relation between them. Let φ : T × X → X , or simply (T, X), be any dynamical system on a space X with a topological semigroup T. We say that x ∈ X is a uniformly recurrent point, almost periodic point of von Neumann, or weakly uniformly recurrent point, if the return time set N(x, U) is syndetic, GH-syndetic, or d F ϕ (N (x , U) > 0 , respectively, where U is a neighborhood of the point x and N (x , U) = { t : t x ∈ U } . It is known that there is no relation between the set of uniformly recurrent points and the set of almost periodic points of von Neumann for a semiflow (T, X). We give further examples for it. We introduce a notion of f-uniformly recurrent point, where f : X → R + is upper semicontinuous and show that x ∈ X is uniformly recurrent if and only if it is an f-uniformly recurrent point for every upper semicontinuous f : X → R + on the regular space X. In the case of metric space X, f : X → R + is a continuous function. Also, x ∈ X is a uniformly recurrent point if and only if x ∈ Ax ¯ for every thick set A of T. Assume that S is a closed normal non-trivial subsemigroup of T. We prove that every uniformly recurrent point of (S, X) is a uniformly recurrent point of (T, X). The converse holds if T is a discrete semigroup. Let (T, X) be a semiflow on topological space X. Then we show that every two nonempty open sets in X share an orbit of a weakly uniformly recurrent point of (T, X) if and only if (T, X) is a topologically transitive with a dense set of weakly uniformly recurrent points. Finally, we give topological version of sensitive dependence on the initial condition for semiflow (T, X) on topological space X and we show that if the semiflow (T, X) is nonminimal and every two non-empty open sets share an orbit of a weakly uniformly recurrent point, then (T, X) is syndetic-sensitive. [ABSTRACT FROM AUTHOR]

Let [0,∞) be the set of all non-negative real numbers. The set B[0,∞) = [0,∞) × [0,∞) with the following binary operation (a, b)(c, d) = (a + c - min{b, c}, b + d - min{b, c}) is a bisimple inverse semigroup. In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup B[0,∞) with an adjoined compact ideal of the following tree types. The semigroup B[0,∞) with the induced usual topology τu from R2, with the topology τL which is generated by the natural partial order on the inverse semigroup B[0,∞), and the discrete topology are denoted by B1 [0,∞), B2 [0,∞), and Bd [0,∞), respectively. We show that if SI 1 (SI 2) is a Hausdorff locally compact semitopological semigroup B1 [0,∞) (B2 [0,∞)) with an adjoined compact ideal I then either I is an open subset of SI 1 (SI 2) or the topological space SI 1 (SI 2) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on S1 0 = B1 [0,∞) ∪ {0} (resp. S2 0 = B2 [0,∞) ∪ {0}) with an adjoined zero 0 is either homeomorphic to the one-point Alexandroff compactification of the topological space B1 [0,∞) (resp. B2 [0,∞)) or zero is an isolated point of S1 0 (resp. S2 0). Also, we proved that if SId is a Hausdorff locally compact semitopological semigroup Bd [0,∞) with an adjoined compact ideal I then I is an open subset of SId. [ABSTRACT FROM AUTHOR]

We study two variants of the cosine subtraction law on a semigroup S. The main objective is to solve g (x y ∗) = g (x) g (y) + f (x) f (y) for unknown functions g , f : S → C , where x ↦ x ∗ is an anti-homomorphic involution. Until now this equation has not been solved on non-commutative semigroups, nor even on non-Abelian groups with x ∗ : = x - 1 . We solve this equation on semigroups under the assumption that g is central, and on groups generated by their squares under the assumption that x ∗ : = x - 1 . In addition we give a new proof for the solution of the variant g (x σ (y)) = g (x) g (y) + f (x) f (y) , where σ : S → S is a homomorphic involution. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some concepts of ★-(quasi)-pseudometric spaces, and give an example which shows that there is a ★-quasi-pseudometric space which is not a quasi-pseudometric space. We also study the conditions under which ★-quasi-pseudometric semitopological groups are paratopological groups or topological groups.

It is known that every pseudocompact topological group is precompact, we extend this result to a class of subsemigroup of topological groups. Then we use this results to prove that cancellative locally compact countably compact topological semigroups with open shifts are topological groups and to give a sufficient condition under which a locally compact monothetic topological semigroup is a compact topological group.

We consider two variants of the sine subtraction law on a semi-group S. The main objective is to solve f(xy∗ ) = f(x)g(y) − g(x)f(y) for unknown functions f, g : S → ℂ, where x ↦ x* is an anti-homomorphic involution. Until now this equation was not solved even when S is a non-Abelian group and x* = x−1. We find the solutions assuming that f is central. A secondary objective is to solve f(xσ(y)) = f(x)g(y) − g(x)f(y), where σ : S → S is a homomorphic involution. Until now this variant was solved assuming that S has an identity element. We also find the continuous solutions of these equations on topological semigroups.

We obtain conditions on fuzzy quasi-pseudometrics on either semigroups or groups which imply that they are either fuzzy topological semigroups or topological groups. Our main results are: (1) Let (S , M , ∗) be a fuzzy quasi-pseudometric right topological semigroup (resp., group) such that (M , ∗) is left weakly invariant; then (S , M , ∗) is a fuzzy quasi-pseudometric topological semigroup (resp., group); (2) Suppose that (M , ∗) is a left weakly invariant fuzzy quasi-pseudometric on a monoid G such that each left translation of G is open and every right translation is continuous at the identity e of (G , M , ∗) ; then (G , M , ∗) is a fuzzy quasi-pseudometric topological semigroup. Many results in Sánchez and Sanchis (Fuzzy Sets Syst 330:79–86, 2018) are improved. We also study complete weakly invariant fuzzy metrics (in the sense of Kramosil and Michálek) on semigroups. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. We solve the cosine–sine functional equation f (x y) = f (x) g (y) + g (x) f (y) + h (x) h (y) for unknown functions f , g , h : S → C . The solutions on a group were found by Chung, Kannappan, and Ng in 1985. More recently the solutions on several large classes of semigroups have been found. Here we give the solutions on a general semigroup. The solutions are expressed in terms of multiplicative functions, the solutions of special cases of the sine addition law with one function multiplicative, and the solutions of special cases of the cosine–sine equation with g multiplicative. This gives a complete description since the solutions of the aforementioned special cases are known. The continuous solutions on topological semigroups are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some concepts of ⋆-(quasi)-pseudometric spaces, and give an example which shows that there is a ⋆-quasipseudometric space which is not a quasi-pseudometric space. We also study the conditions under which ⋆-quasi-pseudometric semitopological groups are paratopological groups or topological groups. [ABSTRACT FROM AUTHOR]

We study the semigroup BFnω, which is introduced in the paper [Visnyk Lviv Univ. Ser. Mech.- Mat. 2020, 90, 5–19 (in Ukrainian)], in the case when the ω-closed family Fn generated by the set {0, 1, . . ., n}. We show that the Green relations D and J coincide in BFnω, the semigroup BFnω is isomorphic to the semigroup In+1 ω (conv −−→) of partial convex order isomorphisms of (ω, 6) of the rank 6 n + 1, and BFnω admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup BFnω . In particular, we prove that for any shift-continuous T1 -topology τ on the semigroup BFnω every non-zero element of BFnω is an isolated point of (BFnω, τ), BFnω admits the unique compact shift-continuous T1 -topology, and every ωd-compact shift-continuous T1 -topology is compact. We describe the closure of the semigroup BFnω in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup BFnω is H-closed in the class of Hausdorff topological semigroups. [ABSTRACT FROM AUTHOR]

We investigate the bounded derivations and bounded crossed homomorphisms from S into X * (the first dual of X). We show that the innerness of these bounded derivations implies that S is inner amenable. We prove that every left (right) crossed homomorphism on a semigroup is principal if and only if it is left (right) amenable. Finally, we show that every bounded left (right) crossed homomorphism from S into M(X), the Banach space of all Borel measures on X, is principal. In the locally compact group case, this is an answer for the derivation problem on locally compact groups. [ABSTRACT FROM AUTHOR]

Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

It is known that every pseudocompact topological group is precompact, we extend this result to a class of subsemigroup of topological groups. Then we use this results to prove that cancellative locally compact countably compact topological semigroups with open shifts are topological groups and to give a sufficient condition under which a locally compact mono- thetic topological semigroup is a compact topological group. [ABSTRACT FROM AUTHOR]

Introduction Algebraic structures endowed with a topology have many applications in pure and applied sciences. Instance of such structures are completely simple semigroups which are the nearest relatives of groups. Many studies have been done on completely simple semigroups (see [3], [4], [6], [9], [10]). Dekany in [4] introduced a group congruence on certain completely simple semigroups, having in mind that such semigroups are indeed Rees matrix semigroups. In this paper, we consider a collection ℵ of normal subgroup with finite intersection property. We define a uniformity on the Rees matrix semigroup. So, we study the topological properties of this uniform topology. In particular, we show that if the normal subgroups have arbitrary intersection property, then the uniformity is complete. Finally, we show that every topological Rees matrix semigroup with normal subgroups have a completion. subsemigroup. Material and methods In this scheme, first we introduce a uniformity on a normal system Rees matrix semigroup S. We obtain some properties of topology induced by uniformity and then we establish the semigroup with this topology is a topological paragroup. After that, we construct a complete topological paragroup that contains S as a dense subsemigroup. Results and discussion If ℵ is a family of normal subgroups of G which is closed under intersection. The topological paragroup (S; ℵ) is compact if and only if (S; ℵ) is totally bounded. Finally, we construct a complete topological paragroup that contains S as a dense subsemigroup. Conclusion The following conclusions were drawn from this research. • We introduce a uniformity on a normal system Rees matrix semigroup S. We obtain some properties of topology induced by uniformity. • We find a relation between two topologies induced by two normal systems of G on S. • We prove that (S; ℵ) is a topological semigroup and topological paragroup. • We construct a complete topological paragroup that contains S as a dense subsemigroup. [ABSTRACT FROM AUTHOR]

Let BFω be the bicyclic semigroup extension for the family F of ω-closed subsets of ω which is introduced in [19]. We study topologizations of the semigroup BFω for the family F of inductive ω-closed subsets of ω. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup [6,12] and show that every Hausdorff shift-continuous topology on the semigroup BFω is discrete and if a Hausdorff semitopological semigroup S contains BFω as a proper dense subsemigroup then S\BFω is an ideal of S. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on BFω with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on BFω with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup BFω⊔ I with an adjoined compact ideal I is either compact or the ideal I is open, which extends many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup BFω. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let (X , d ⋆) be a complete ⋆-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of d ⋆ on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d ⋆ is invariant on X. (2) Let (G , d ⋆) be a ⋆-metric group such that d ⋆ is invariant on G. Then, (G , d ⋆) is complete if and only if (G , τ d ⋆) is Raǐkov complete. [ABSTRACT FROM AUTHOR]

It is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero C ℤ 0 is discrete. At the same time, on C ℤ 0 , there exist 픠 different Hausdorff locally compact shift-continuous topologies. In addition, on C ℤ 0 , we construct a unique minimal shift-continuous topology and a unique minimal inverse semigroup topology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup S. The set of all quasi-multipliers on S is denoted by (S). First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if S is a topological semigroup such that S × S is pseudocompact, then (S) can be regarded as a subset of (β S). Moreover, with an extra condition we describe (S) as a quotient subsemigroup of β S. Finally, we investigate quasi-multipliers on topological semigroups, its relationship with multipliers and give some concrete examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish the necessary and sufficient condition for an equivalence relation ρ on an S-act A endowed with a topology such that A=ρ becomes a Hausdorff topological S-act. Also, we show that if A1 and A2 be two topological S-acts, then for any homomorphism φ → A1 ! A2, A1= ker φ is a topological S-act if and only if φ is φ-saturated continuous. Moreover, we establish for any two congruences θ1 and θ2 on an S-act A endowed with a topology, θ1∩θ2 is a topological S-act congruence on A if and only if the mapping φ: A → A/θ1 × A/θ2, defined by φ(a) = (aθ1; aθ2), for all a ∈ A, is φ-saturated continuous, where S is a topological semigroup. [ABSTRACT FROM AUTHOR]

For any affine semigroup S, the set S ∪ { ∞ } has a natural semigroup structure; additionally, if S is endowed with the discrete topology, then the semigroup S ∪ { ∞ } can be studied as the one-point compactification of S. In this article, we study the derivations on semigroup algebra C [ S ∪ { ∞ } ] in relation to the derivations on semigroup algebra C [ S ] considering the metrizable topology on C [ S ∪ { ∞ } ] induced by the one-point compactification topology of S ∪ { ∞ }. [ABSTRACT FROM AUTHOR]

In an earlier paper we showed that spectral synthesis holds for commutative monoids with no prime ideals. We also found that spectral synthesis sometimes holds and sometimes fails for monoids with prime ideals. Here we continue our investigation of Levi-Civita functional equations on commutative semigroups. In particular we try to decide whether spectral synthesis holds if the semigroup has a prime ideal, and we solve Levi-Civita equations in some cases where spectral synthesis fails. We also present some examples showing that the existence of an identity element is not necessary for spectral synthesis. [ABSTRACT FROM AUTHOR]

The symmetric inverse semigroup I(X) on a set X is the collection of all partial bijections between subsets of X with composition as the algebraic operation. We study the minimal Hausdorff inverse semigroup topology on I(X). We present some characterizations of it. When X is countable such topology is Polish. [ABSTRACT FROM AUTHOR]

The cosine addition formula on a semigroup S is the functional equation g (x y) = g (x) g (y) - f (x) f (y) for all x , y ∈ S . We find its general solution for g , f : S → C , using the recently found general solution of the sine addition formula f (x y) = f (x) g (y) + g (x) f (y) on semigroups. A simpler proof of this latter result is also included, with some details added to the solution. We also solve the cosine subtraction formula g (x σ (y)) = g (x) g (y) + f (x) f (y) on monoids, where σ is an automorphic involution. The solutions of these functional equations are described mostly in terms of additive and multiplicative functions, but for some semigroups there exist points where f and/or g can take arbitrary values. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

We show, contrary to some published statements, that spectral synthesis does not generally hold for commutative semigroups that are not groups. On the positive side we prove that it holds if the semigroup is a monoid with no prime ideal. For semigroups with a prime ideal, the picture is not so clear. On the negative side we provide a variety of examples illustrating the failure of spectral synthesis for many semigroups with prime ideals, but we also give examples of semigroups with prime ideals on which spectral synthesis holds. [ABSTRACT FROM AUTHOR]

In this article, we establish necessary and sufficient condition on a topological Clifford semigroup to be a semilattice of topological groups. As a consequence, we show that a topological Clifford semigroup (S , τ) satisfies the property that for each G ∈ τ and every x ∈ G , there exists an element U ∈ τ such that x ∈ U ⊆ G ∩ J x if and only if it is a strong semilattice of topological groups if and only if it is a semilattice of topological groups. We prove that some topological properties like T 0 , T 1 , T 2 , regularity and completely regularity are equivalent in a semilattice of topological groups. We also prove that the quotient space of a semilattice of topological groups by a full normal Clifford subsemigroup is again a semilattice of topological groups. Finally, we establish that if { S i : i = 1 , 2 , ... , n } is a family of semilattices of topological groups and Ni is a full normal Clifford subsemigroup of Si for all i = 1 , 2 , ... , n , then ⊗ i = 1 n (S i / N i) is topologically isomorphic to ⊗ i = 1 n S i / ⊗ i = 1 n N i. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let (X,d⋆) be a complete ⋆-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of d⋆ on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and d⋆ is invariant on X. (2) Let (G,d⋆) be a ⋆-metric group such that d⋆ is invariant on G. Then, (G,d⋆) is complete if and only if (G,τd⋆) is Raǐkov complete.

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group. [ABSTRACT FROM AUTHOR]

The sine addition formula on a semigroup S is the functional equation f (x y) = f (x) g (y) + g (x) f (y) for all x , y ∈ S . For some time the solutions have been known on groups, regular semigroups, and semigroups which are generated by their squares. The obstacle to finding the solution on all semigroups arose in the special case that g is a multiplicative function. We overcome this obstacle and find the general solution on all semigroups using a transfinite induction argument. A new type of solution appears which is not seen on regular semigroups or semigroups generated by their squares. We also give the general solution of the sine subtraction formula f (x σ (y)) = f (x) g (y) - g (x) f (y) on monoids, where σ is an automorphic involution. The solutions of both equations can be described in terms of additive and multiplicative functions, with a slight new twist. The general continuous solutions on topological semigroups are also found. A variety of examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.

Let S be a foundation topological semigroup and M a (S) the space of all measures μ ∈ M (S) for which the maps x ⟼ | μ | ∗ δ x and x ⟼ δ x ∗ | μ | from S into M(S) are weakly continuous. In the present paper, we introduce and study the concept of ϕ -amenability for S and investigate the relations between ϕ -amenability of S and essential ϕ ^ -amenability of M a (S) , where ϕ is a character on S and ϕ ^ is the extension of ϕ to M a (S) . [ABSTRACT FROM AUTHOR]

The sine and cosine addition laws on a (not necessarily commutative) semigroup are f (x y) = f (x) g (y) + g (x) f (y) , respectively g (x y) = g (x) g (y) - f (x) f (y) . Both of these have been solved on groups, and the first one has been solved on semigroups generated by their squares. Quite a few variants and extensions with more unknown functions and/or additional terms have also been studied. Here we extend these results and solve the Levi–Civita functional equation f (x y) = g 1 (x) h 1 (y) + g 2 (x) h 2 (y) by elementary methods on groups and monoids generated by their squares, assuming that f is central. We also find the continuous solutions in the case of topological groups and monoids. [ABSTRACT FROM AUTHOR]

In this paper we study and develop the retracting property in homotopy theory for finite topological semigroups by introducing the notions of DS-retract sets and U S-retract sets. We show that these sets are stronger forms of strong S -deformation retract sets. Furthermore, we study the uniqueness of retraction maps and some topological properties for these retract sets. [ABSTRACT FROM AUTHOR]

We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω- semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological O-bisimple inverse ω-semigroup with a compact maximal subgroup is either compact or it is a topological sum of its H-classes. We describe the structure of Hausdorff locally compact semitopological O-bisimple inverse ω-semigroups with a monothetic maximal subgroups. We show the following dichotomy: a T1 locally compact semitopological Reilly semigroup (B(Z+, θ)0, τ) over the additive group of integers Z+, with adjoined zero and with a non-annihilating homomorphism is either compact or discrete. At the end we establish some properties of the remainder of the closure of the discrete Reilly semigroup B(Z+, θ) in a semitopological semigroup.

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.

Let G be a locally compact first countable Abelian topological group of cardinality ≤ c and suppose that for every n ∈ N the subgroup nG is not totally bounded. We show that (1) the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ (⊕ λ Z +) for some λ ∈ [ p , c ] such that G is dense in T and (S , T) is absolutely closed in the class of cancellative topological semigroups with the Fréchet-Urysohn property, and (2) assuming Martin's Axiom, the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ (⊕ c Z +) such that G is dense in T and (S , T) is absolutely closed in the class of all topological semigroups with the Fréchet-Urysohn property. [ABSTRACT FROM AUTHOR]

In this paper we study submonoids of the monoid j↗∞(N) of almost monotone injective cofinite partial selfmaps of positive integers N. Let j↗∞(N) be a submonoid of j↗∞(N) which consists of cofinite monotone partial bijections of N and b N be a subsemigroup of j↗∞(N) which is generated by the partial shift n 7→ n + 1 and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of j↗∞(N) which contains the semigroup bN is the identity map. We construct a submonoid IN [1] I ∞ (N) with the following property: if S is an inverse submonoid of j↗∞(N) such that S contains IN [1] ∞ as a submonoid, then every non-identity congruence C on S is a group congruence. We show that if S is an inverse submonoid of j↗∞(N) such that S contains bN as a submonoid then S is simple and the quotient semigroup S/Cmg, where Cmg is the minimum group congruence on S, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of j↗∞(N) which contain bN and embeddings of such semigroups into compact-like topological semigroups. [ABSTRACT FROM AUTHOR]

We show that an Abelian topological group G is absolutely closed in the class of topological semigroups if and only if G is complete and there is n∈N such that the subgroup nG={nx:x∈G} is totally bounded. If for every n∈N, the subgroup nG is not totally bounded, then the topology of G can be extended to a semigroup topology T on G×Z+ in which G is open and dense, and if G is locally compact, so can be chosen T. In particular, the topology of R can be extended to a locally compact semigroup topology on R×Z+ in which R is dense. We also show that the topology of G can be extended to a regular (equivalently, Tychonoff) semigroup topology on G×Z+ in which G is open and dense if and only if there is a neighborhood U of 0∈G such that for every n∈N, the subgroup nG is U-unbounded. [ABSTRACT FROM AUTHOR]

In this paper we generalize the retracting property in homotopy theory for topological semigroups by introducing the notions of deformation S-retraction with its weaker forms and ES-homotopy extension property. Furthermore, the covering homotopy theorems for S-maps into Sχ-fibrations and Sχ-cofibrations are introduced and pullbacks for Sχ-fibrations behave properly.