Dense Subsemigroup Research Articles

Force recurrence near zero

Sets that enjoy with recurrence properties near zero, in particular, sets that force recurrence near zero, sets that contain broken IP-set near zero, sets that force uniform recurrence near zero and sets that contain broken syndetic set near zero have been introduced and some relations among them are established on countable dense subsemigroup of the additive semigroup of positive real numbers. The article begins with a characterization of recurrent point near zero that affects the results exhibited here. It is also showed that how IP-set near zero can be realized dynamically.

On a semitopological semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ when a family $\mathscr{F}$ consists of inductive non-empty subsets of $\omega$

Let $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ be the bicyclic semigroup extension for the family $\mathscr{F}$ of ${\omega}$-closed subsets of $\omega$ which is introduced in \cite{Gutik-Mykhalenych=2020}.We study topologizations of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ for the family $\mathscr{F}$ of inductive ${\omega}$-closed subsets of $\omega$. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup \cite{Bertman-West-1976, Eberhart-Selden=1969} and show that every Hausdorff shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is discrete and if a Hausdorff semitopological semigroup $S$ contains $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ as a proper dense subsemigroup then $S\setminus\boldsymbol{B}_{\omega}^{\mathscr{F}}$ is an ideal of $S$. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ 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 $\boldsymbol{B}_{\omega}^{\mathscr{F}}$ with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}\sqcup I$ with an adjoined compact ideal $I$ is either compact or the ideal $I$ is open, which extent many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}}$.

Partition regularity of polynomial systems near zero

Recently, S. Kanti Patra and Md. Moid Shaikh proved the existence of monochromatic solutions to systems of polynomial equations near zero for particular dense subsemigroups (S,+) of (mathbb {R}^{+},+). We extend their results to a much larger class of systems whilst weakening the requests on S, using solely basic results about ultrafilters.

Dynamics near an idempotent

Hindman and Leader first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of ((0,∞),+). Using the algebraic structure of the Stone-Čech compactification, Tootkaboni and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent e for a dense subsemigroup of a semitopological semigroup (R,+) and they gave the combinatorial proof of the Central Sets Theorem near e. Algebraically one can define quasi-central sets near e for dense subsemigroups of (R,+). In a dense subsemigroup of (R,+), C-sets near e are the sets, which satisfy the conclusions of the Central Sets Theorem near e. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near e. We also study minimal systems near e in the last section of this paper.

Combined algebraic properties of $$C^{*}$$-sets near zero

It is known that for an $$IP^*$$ set A in $${\mathbb {N}}$$ and a sequence $$\langle {x_n}\rangle _{n=1}^{\infty }$$ there exists a sum subsystem $$\langle {y_n}\rangle _{n=1}^{\infty }$$ of $$\langle {x_n}\rangle _{n=1}^{\infty }$$ such that $$FS\left( \langle {y_n}\rangle _{n=1}^{\infty }\right) $$$$\cup $$$$FP\left( \langle {y_n}\rangle _{n=1}^{\infty }\right) \subseteq A$$. Similar types of results also have been proved for central$$^*$$ and $$C^*$$-sets where the sequences considered belong to a restricted class of sequences. In 2012, D. De and R. K. Paul have extended these results for $$IP^*$$ and central$$^*$$ sets near zero in dense subsemigroups of $$((0,\infty ),+)$$. In this present work we will extend the results for $$C^*$$-sets near zero in dense subsemigroups of $$((0,\infty ),+)$$.

Dynamical characterizations of combinatorially rich sets near zero

Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of ((0,∞),+) and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone–Čech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central sets theorem near zero. Algebraically one can define quasi-central sets near zero for dense subsemigroups of ((0,∞),+), and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of ((0,∞),+), C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. We shall produce dynamical characterizations of these combinatorially rich sets near zero.

Central Sets Theorem near zero

In this paper, we introduce notions of J-set near zero and C-set near zero for a dense subsemigroup of ((0,+∞),+) and state the Central Sets Theorem near zero. Among the other results for a dense subsemigroup S⊆((0,+∞),+), we give some sufficient and equivalent algebraic conditions on a subset A⊂S to be a J-set near zero and to be a C-set near zero.

Density near zero

Let $S$ be a dense subsemigroup of $(0,+infty)$. In this paper, we state definition of thick near zero, and also we will introduce a definition that is equivalent to the definition of piecewise syndetic near zero which presented by Hindman and Leader in [6]. We define density near zero for subsets of $S$ by a collection of nonempty finite subsets of $S$ and we investigate the conditions under these concepts.

There is no domain representable dense proper subsemigroup of a topological group

We prove that the only domain representable dense subsemigroup of a topological group is itself. Consequently, we obtain that every domain representable subgroup of a topological group is closed.

Central Sets Theorem near zero

In this paper, we introduce notions of J-set near zero and C-set near zero for a dense subsemigroup of ((0,+∞),+) and state the Central Sets Theorem near zero. Among the other results for a dense subsemigroup S⊆((0,+∞),+), we give some sufficient and equivalent algebraic conditions on a subset A⊂S to be J-set near zero and to be C-set near zero.

SUBSEMIGROUPS OF SEMISIMPLE LIE GROUPS

Let G be a simple Lie group with finite center. We show that G can be algebraically generated as a semigroup by a one-parameter subsemigroup X + and one additional element g. In fact, given one of the two, a non-constant X + or a non-central g, there is a g, respectively X +, such that the two together generate G as a semigroup. It follows that given a non-constant one-parameter subsemigroup X + of G there is another one-parameter subsemigroup Y + such that the two generate G as a semigroup. Similarly, given a non-central element g of G there is an element h of G such that the two generate a dense subsemigroup of G. We have analogous results for semismple Lie groups. For SL(2, ℝ) and its universal cover we spell out when two one-parameter subsemigroups generate the group as a semigroup. On the way we prove results on open subsets of subsemigroups and on exponentiality, which may be of independent interest.

The semigroup of ultrafilters near an idempotent of a semitopological semigroup

Let (T,+) be a Hausdorff semitopological semigroup, S be a dense subsemigroup of T and e be an idempotent element of T. The set eS⁎ of ultrafilters on S that converge to e is a semigroup under restriction of the usual operation + on βTd, the Stone–Čech compactification of the discrete semigroup Td. We characterize the smallest ideal of (eS⁎,+), and those sets “central” in (eS⁎,+), that is, those sets which are members of minimal idempotents in (eS⁎,+). We describe some combinatorial applications of those sets that are central in (eS⁎,+).

Combined Algebraic Properties of IP* and Central* Sets Near 0

It is known that for an IP* setAinℕand a sequence〈xn〉n=1∞there exists a sum subsystem〈yn〉n=1∞of〈xn〉n=1∞such thatFS(〈yn〉n=1∞) ∪ FP(〈yn〉n=1∞)⊆A. Similar types of results also have been proved for central* sets. In this present work we will extend the results for dense subsemigroups of((0,∞),+).

Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices

In this paper we bring together results about the density of subsemigroups of abelian Lie groups, the minimal number of topological generators of abelian Lie groups and a result about actions of algebraic groups. We find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup (or subgroup) of the connected component of the identity of its Zariski closure.

Dense 2-generator subsemigroups of [formula omitted] matrices

We construct explicit examples of 2-generator dense subsemigroups of GL ( 2 , R ) and SL ( 2 , R ) . In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups of GL ( 2 , C ) . We also show that the semigroup of real linear fractional transformations on a proper subinterval of the real line does not admit any 2-generator dense subsemigroups, and then we construct a 3-parameter family of examples of 3-generator dense subsemigroups.

Mesures stationnaires et fermés invariants des espaces homogènes II

Let G be a real Lie group, Λ be a lattice of G, H be a connected semisimple subgroup of G with no compact factor and Γ be a Zariski dense sub-semigroup of H. We prove that every Γ-orbit closure in the quotient space X = G / Λ is homogeneous. Let μ be a probability measure on G whose support is compact and spans Γ. We prove that every μ-stationary μ-ergodic probability measure on X is Γ-invariant and homogeneous.

Generating semisimple groups by tori

Let G be a simple real Lie group with finite center. We prove that given a non-central element g in G there is an elliptic element h in G such that h and g h g − 1 generate a dense subsemigroup of G.

Extending binary operations to funtor-spaces

Given a continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subsemigroup of all elements of TX that have finite support.

Free subsemigroups in topological semigroups

We prove that a complete metrizable topological semigroup with identity that contains a free dense subsemigroup and no commutative neighbourhoods of identity is almost free. We apply this result to the full transformation semigroup of countable degree and the semigroup of automaton transformations over finite alphabet.

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}.