In this paper, we have introduced the notion of (1,2)-ideal in an LA-semigroup and shown that (1,2)-ideal and two-sided ideal coincide in an intra-regular LA-semigroup. We have characterized an intra-regular LA-semigroup by using the properties of left and right ideals. Some natural examples of LA-semigroups have been given. Further we have investigated some useful conditions for an LA-semigroup to become an intra-regular LA-semigroup and given the counter examples to illustrate the converse inclusions. All the ideals (left, right, two-sided, interior, quasi, bi- generalized bi- and (1,2)) of an intra-regular LA-semigroup have been characterized. Finally we have given an equivalent statement for a two-sided ideal of an intra-regular LA-semigroup in terms of the intersection of two minimal two-sided ideals of an intra-regular LA-semigroup.
We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable i…
Ideal series of semigroups play an important role in the examination of semigroups which have proper two-sided ideals. But the corresponding theorems cannot be used when left simple (or right simple or simple) semigroups are considered. So it is a natural idea that we try to use the group theoretical methods (instead of the ring theoretical ones) for the examination of these semigroups. The purpose of this paper is to find a suitable type of subsemigroups of left simple semigroups which makes possible for us to generalize some notions (the notion of the normal series and the composition series…
Following G.Szasz [2] a subsemigroup I of semigroup S is called an interior ideal if SIS \subset I. In this paper we explore the classes of regular semigroup and its different subclasses by their interior ideals. Furthermore, we introduce the strongly prime, prime, semiprime, strongly irreducible, and irreducible interior ideals of semigroups and also characterize those semigroups for which each interior ideal is strongly prime. Some important interplay between the classes of all interior ideals and other ideals are given here. In addition to this, we present different characterizations of sem…
In this paper 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\in {\mathbb F}[S]:\ {\mathbb F}[S…
In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.
Let $M(S)$ be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroup $S$ with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions for $M(S)^{*}$ to have a topologically left invariant mean.
A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered un…
The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products o…
A partial semigroup is a set with restricted binary operation. In this work we will extend a result due to V. Bergelson and N. Hindman concerning the rich structure presented in the product space of semigroups to partial semigroup. An $IP^{\star}$ set in a semigroup is a set that intersect every set of the form $\left\{ FS(x_{n})_{n=1}^{\infty}:x_{n}\in S\right\} $. V. Bergelson and N. Hindman proved that if $S_{1},S_{2},\ldots,S_{l}$ are finite collection of commutative semigroup, then under certain condition, an $IP^{\star}$ set in $S_{1}\times S_{2}\times\ldots\times S_{l}$ contains cartesi…
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{T}_{n}$ be the semigroup of full transformations on $[n]$. Let $\mathcal{CT}_{n}=\{α\in \mathcal{T}_{n}: (for ~all~x,y\in [n])~\left|xα-yα\right|\leq\left|x-y\right|\}$, then $\mathcal{CT}_{n}$ is a subsemigroup of $\mathcal{T}_{n}$. In this paper, we give a necessary and sufficient condition for an element to be regular and characterize all the Green's equivalences for the semigroup $\mathcal{CT}_{n}$. We further show that the semigroup $\mathcal{CT}_{n}$ is a left abundant semigroup.
In this paper 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 contains 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 characterize finite semilattice indecomposable semigroups with a zero by only use the properties of its semigroup algebra.
In this article we study left I-orders in the bicyclic monoid $\mathcal{B}$. We give necessary and sufficient conditions for a subsemigroup of $\mathcal{B}$ to be a left I-oreder in $\mathcal{B}$. We then prove that any left I-order in $\mathcal{B}$ is straight.
A subsemigroup $S$ of an inverse semigroup $Q$ is a left I-order in $Q$, if every element in $Q$ can be written as $a^{-1}b$ where $a, b \in S$ and $a^{-1}$ is the inverse of $a$ in the sense of inverse semigroup theory. We study a characterisation of semigroups which have a primitive inverse semigroup of left I-quotients.
We introduce two natural notions of cogrowth for finitely generated semigroups --- one local and one global --- and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when d…
It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal sub…
Given a semigroup S with zero, which is left-cancellative in the sense that st=sr \neq 0 implies that t=r, we construct an inverse semigroup called the inverse hull of S, denoted H(S). When S admits least common multiples, in a precise sense defined below, we study the idempotent semilattice of H(S), with a focus on its spectrum. When S arises as the language semigroup for a subsift X on a finite alphabet, we discuss the relationship between H(S) and several C*-algebras associated to X appearing in the literature.
In this paper, the notion of bipolar-valued fuzzy LA-subsemigroups is introduced and also some properties of bipolar-valued fuzzy left (right, bi-, interior) ideals of LA-semigroups has been discussed.
In this paper, we characterize intra-regular LA-semihypergroups by using the properties of their left and right hyperideals and we investigate some useful conditions for an LA-semihypergroup to become an intra-regular LA-semihypergroup.
In the paper we introduce a notion of the Bruck-Reilly $λ$-polycyclic extension of a monoid $S$ with a homomorphism $θ$ which is an analogue of the Bruck-Reilly extension of a monoid $S$. We describe idempotens of the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ and Green's relations on $\left(\mathscr{P}_λ(θ,S),*\right)$. It is proved that $\left(\mathscr{P}_λ(θ,S),*\right)$ is a $0$-simple semigroup for any semigroup $S$. We find necessary and sufficient conditions on a monoid $S$ and a homomorphism $θ$ under which the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ is regular, inverse, $0$-b…
When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The…
Let $M$ be a cancellative monoid. It is known~\cite{Ta54} that if $M$ is left amenable then the monoid ring $K[M]$ satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. In~\cite{Don10} Donnelly shows that a partial converse to this statement is true. Namely, if the monoid $\mathbb Z^{+}[M]$ of all elements of $\mathbb Z[M]$ with positive coefficients has nonzero common right multiples, then $M$ is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of gro…
For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) $\le$ W (S), we show that if |L| $\le$ 12 then W 0 (S) $\ge$ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilf's conjecture remains open for |L| $\ge$ 13.
Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $Sing(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathscr{P}(X)$ and the category of partitions $Π(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further we prove that every cross-connection between them …
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
In this paper we characterize left(right) ideals, bi-ideals and quasi-ideals of an ordered semigroup by an index $m$ and give some important interplays between these ideals. The concept of m-regularity of an ordered semigroups has been introduced. Moreover m-regular ordered semigroups are characterized by their m-quasi-ideals and the fact that for any m-regular ordered semigroups $A$, the set $Q_{A}$ of all m-quasi-ideals of $A$, with multiplication defined by: $Q_{1}\circ Q_{2}=(Q_{1}Q_{2}]$, for all $Q_{1},Q_{2}\in Q_{A}$, is a m-regular semigroup is obtained here.
On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence also to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.
The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.
We consider the general question of how the homological finiteness property left-FPn holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. In particular we show that left-FPn is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups we prove the converse, and as a corollary show that a completely simple semigroup is of type left- and right-FPn if and only if it has finitely many left and right ideals and all of its m…
We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \[G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{left commutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $Λ(G),$ are the semigroups of mappings generated by $ρ(g):G \to G$ and $λ(g):G \to G$ defined by $(x)ρ(g) = …
We call a semigroup $S$ right noetherian if it satisfies the ascending chain condition on right ideals, and we say that $S$ satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal $λ$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_λ^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ the following conditions are equivalent: $(i)$ $τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d…
Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $τ\colon A^S \to A^S$ is surjective.
In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $\ell^1_+(S)$ have a common nonzero right multiple, where $\ell^1_+(S)$ is the positive part of the Banach algebra $\ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.
We provide an infinite family of left-ordered groups, all of which have a positive cone that is finitely generated as a semigroup. This family includes the Klein bottle group and the braid group B_3.
