Proper restriction semigroups – semidirect products and W-products

Fountain and Gomes [4] have shown that any proper left ample semigroup embeds into a so-called W-product, which is a subsemigroup of a reverse semidirect product ${T\ltimes {\mathcal {Y}}}$ of a semilattice ${\mathcal {Y}}$ by a monoid T, where the action of T on  ${\mathcal {Y}}$ is injective with images of the action being order ideals of  ${\mathcal {Y}}$ . Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim of this paper is to give necessary and sufficient conditions on a proper left restriction semigroup such that it embeds into a W-product. We also examine the complex relationship between W-products and semidirect products of the form ${{\mathcal {Y}}\rtimes T}$ .     

Finite derivation type for semilattices of semigroups

In this paper we investigate how the combinatorial property finite derivation type (FDT) is preserved in a semilattice of semigroups. We prove that if $S= \mathcal{S}[Y,S_{\alpha}]$ is a semilattice of semigroups such that Y is finite and each S _?? (????Y) has FDT, then S has FDT. As a consequence we can show that a strong semilattice of semigroups $\mathcal{S}[Y,S_{\alpha},\lambda_{\alpha,\beta}]$ has FDT if and only if Y is finite and every semigroup S _?? (????Y) has FDT.     

Left Equalizer Simple Semigroups

We characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence \({\varrho}\) on a semigroup S, let \({{\mathbb F}[\varrho]}\) denote the ideal of the semigroup algebra \({{\mathbb F}[S]}\) which determines the kernel of the extended homomorphism of \({{\mathbb F}[S]}\) onto \({{\mathbb F}[S/\varrho]}\) induced by the canonical homomorphism of S onto \({S/\varrho}\). We examine the right colons (\({{\mathbb F}[\varrho] :_{r} {\mathbb F}[S]) = {a \epsilon {\mathbb F}[S] : {\mathbb F}[S]a \subseteqq {\mathbb F}[\varrho]}}\) in general, and in that special case when \({\varrho}\) has the property that the factor semigroup \({S/\varrho}\) is left equalizer simple.     

Smoothability and order bound for AS semigroups

In this paper we consider numerical semigroups S generated by arithmetic sequences m ₀,??,m _n (AS-semigroups). First we state some results on the module $T^{1}_{k[S]}$ ; further in the cases m ₀??1 and m ₀??n (modulo n), we prove these semigroups are Weierstrass by showing that the associated monomial curves $X=\operatorname {Spec}{k[S]}$ are smoothable. Finally for each semigroup S generated by an arithmetic sequence we evaluate the so-called ??order bounds??: when S is Weierstrass, these invariants are good approximations for the minimum distance of the related one-point codes.     

Необходимое условие полноты системы экспонент в прострaнстве квaдрaтично интегрируемых функций со степенным весом

A necessary condition is obtained for the completeness of the system of exponents $$e(\Lambda ) = \left\{ {e^{ - \lambda _n t} :\lambda _n \in \left\{ {z:0 < \operatorname{Re} z < A \in \mathbb{R}^ + ,0 < \operatorname{Im} z < 2\pi } \right\}} \right\}$$ in the space of square integrable functions with the power weight t ^?? , where ?1 < ?? < 0.     

Semilattice indecomposable finite semigroups with large subsemilattices

We show that if Y is a subsemilattice of a finite semilattice indecomposable semigroup S then \({|Y|\leq 2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1}\). We also characterize finite semilattice indecomposable semigroups S which contain a subsemilattice Y with \({|S|=4k+1}\) and \({|Y|=2\left\lfloor \frac{|S|-1}{4} \right\rfloor+1=2k+1}\). They are special inverse semigroups. Our investigation is based on our new result proved in this paper which characterizes finite semilattice indecomposable semigroups with a zero by using only the properties of its semigroup algebra.     

Algebraic properties of Zappa–Szép products of semigroups and monoids

Direct, semidirect and Zappa–Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa–Szép products of monoids and semigroups and investigate generalised Greens relations \({\mathcal R}^{*},\, {\mathcal L}^{*},\, \widetilde{\mathcal {R}}_E\) and \(\widetilde{\mathcal {L}}_E\) for these Zappa–Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa–Szép product \(E \bowtie S\). We further investigate properties of \(E \bowtie S\) and show that S is a retract of \(E\bowtie S\). We also find a subset T of \(E \bowtie S\) which is left restriction.     

Almost periodic compactifications of semidirect products of flows

Let $S\tilde \times T$ be a semidirect product of semitopological semigroups S and T. If S and T act on topological spaces X and Y, respectively, then under suitable conditions there is a natural action of $S\tilde \times T$ on X × Y. In this paper we characterize the almost periodic and strongly almost periodic compactification of the flow ( $S\tilde \times T$ , X × Y) in terms of related compactifications of (S, X) and (T, Y).     

A note on maximal regular subsemigroups of the finite transformation semigroups \mathcal{T}(n,r)

Let $\mathcal{T}_{n}$ be the semigroup of all full transformations on the finite set X _n ={1,2,…,n}. For 1≤r≤n, set $\mathcal {T}(n, r)=\{ \alpha\in\mathcal{T}_{n} | \operatorname{rank}(\alpha)\leq r\}$ . In this note we show that, for 2≤r≤n?2, any maximal regular subsemigroup of the semigroup $\mathcal{T} (n,r)$ is idempotent generated, but this may not happen in the semigroup $\mathcal{T}(n, n-1)$ .     

The cardinality of β A (S J )

Let J be an infinite set and let $I=\mathcal{P}_{f}( J)$ . For i??I, define $\mathcal{B}_{J}( i) =\{ f\mid f:\mathcal{P}( i) \rightarrow \mathcal{P}( i) \} $ and let $$S_{J}=\{ ( i,f) \mid i\in I\text{ and } f\in \mathcal{B}_{J}( i) \}.$$ For (i,f), (k,g)??S _J , define $f\ast g:\mathcal{P}( i\cup k) \rightarrow \mathcal{P}( i\cup k) $ as follows. For $x\in \mathcal{P}( i\cup k) $ , let $$( f\ast g) ( x) =\left\{\begin{array}{l@{\quad }l}g( x) , & \text{if\ }x=\emptyset, \\g( x\cap k) , & \text{if\ }x\cap k\neq \emptyset, \\f( x) , & \text{if\ }x\in \mathcal{P}( i\backslash k)\text{ and }x\neq \emptyset.\end{array}\right.$$ Define (i,f)?(k,g)=(i??k,f?g). It is shown that (S _J ,?) is a semigroup. Let ??S _J denote the collection of all ultrafilters on the set S _J . We consider (??S _J ,?), the compact (Hausdorff) right topological semigroup that is the Stone?C?ech Compactification of the semigroup (S _J ,?) equipped with the discrete topology. Similar to the construction in Grainger (Semigroup Forum 73:234?C242, 2006), it is shown that there is an injective map A???? _A (S _J ) of $\mathcal{P}( J) $ into $\mathcal{P}( \beta S_{J}) $ such that each ?? _A (S _J ) is a closed subsemigroup of (??S _J ,?), the set ?? _J (S _J ) is the smallest ideal of (??S _J ,?) and the collection $\{ \beta_{A}( S_{J}) \mid A\in \mathcal{P}( J) \} $ is a partition of???S _J . The main result is establishing that the cardinality of??? _A (S _J ) is $2^{2^{\vert J\vert }}$ for any?A?J.     

Regularity of semigroups via the asymptotic behaviour at zero

An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) _t≥0 on a uniformly convex space X is holomorphic iff $\limsup_{t \downarrow0} \|T(t) - \operatorname{Id}\| < 2$ . We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent θ∈(0,1). As an application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to $\mathcal{R}$ -sectoriality: We obtain a characterization of maximal regularity by the behaviour of the semigroup at zero and show extrapolation results.     

Left Orders in Special Completely 0-Simple Semigroups

A subsemigroup S of a semigroup Q is a left order in Q, and Q is a semigroup of left quotients of S, if every element of Q can be written as a ^?1 b for some ${a, b\in S}$ with a belonging to a group ${\mathcal{H}}$ -class of Q. Characterizations are provided for semigroups which are left orders in completely 0-simple semigroups in the following classes: without similar ${\mathcal{L}}$ -classes, without contractions, ${\mathcal{R}}$ -unipotent, Brandt semigroups and their generalization. Complete discussion of two examples and an idea for a new concept conclude the paper.     

Infinite compact sets of idempotents in $$\beta S$$

We show that if S is a countably infinite right cancellative semigroup and T is an infinite compact set of idempotents in the Stone–?ech compactification \(\beta S\) of S, then T contains an infinite compact left zero semigroup.     

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ??, ??, holding on suitable dilation invariant supersets of C ₀ ^?? (?₊). Maximal sets of admissible functions u are described. This paper is based on authors?? earlier abstract results and applies them to particular classes of weights.     

On the tolerance lattice of tolerance factors

Although the notion of a tolerance is a natural generalization of the notion of a congruence, many properties of factor lattices modulo congruences are not, in general, valid for factor lattices modulo tolerances. In this paper, for a lattice L of a finite length, we define a new partial order ? on $\operatorname{Tol}\, (L)$ such that for every ${S\in \operatorname{Tol}\, (L)}$ with T?S, a tolerance S/T is induced on the factor lattice L/T. This partial order is a particular restriction of ? and thus we can prove for tolerances some analogous results to the homomorphism theorem and the second isomorphism theorem for congruences. The poset $(\operatorname{Tol}\, (L), \sqsubseteq)$ is not always a lattice, but it can be converted into a specific commutative join-directoid. Then, for every ${T\in \operatorname{Tol}\, (L)}$ , $(\operatorname{Tol}\, (L/T),\sqsubseteq)$ constitutes a subdirectoid of the directoid based on the poset $(\operatorname{Tol}\, (L),\sqsubseteq)$ and this specific directoid structure is preserved by the direct product of lattices.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .     

Rank and idempotent rank of finite full transformation semigroups with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. In 2011, Sanwong studied the regular part $$F(X,Y)=\bigl\{\alpha\in T(X,Y): X\alpha\subseteq Y\alpha\bigr\}, $$ of T(X,Y) and described its Green’s relations and ideals. In this paper, we compute the rank of F(X,Y) when X is a finite set. Moreover, we obtain the rank and idempotent rank of its ideals.