We investigate semigroups $S$ which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on $S$. We call such $S$ a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups -- including periodic, stable, inverse and several well-known types of simple semigroups -- the only DSC members are groups. However, it turns out that there exist non-group DSC semigroups, which we obtain utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.
We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X=N of natural numbers containing a given subsemigroup W of T(X) where each element of a given set $U$ is a generator of T(X) modulo W. This note continue the study of maximal subsemigroups on the monoid of all full transformations on a infinite set.
We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.
All isolated, completely isolated, and nilpotent subsemigroups in the semigroup $\IS$ of all injective partial transformations of an $n$-element set, considered as a semigroup with a sandwich multiplication are described.
Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by $x\star_ay=xay$ for $x,y\in S$. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semi…
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently co…
We prove new results on inheritance of Green's relations by subsemigroups in the presence of stability of elements. We provide counterexamples in other cases to show in particular that not all right-stable semigroups are embeddable in left-stable semigroups. This is carried out in the context of a survey of the various closely related notions of stability and minimality of Green's classes that have appeared in the literature over the last sixty years, and which have sometimes been presented in different forms.
In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the infinite set $Ω$, containing certain subgroups of the symmetric group $\sym(Ω)$ on $Ω$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing $\sym(Ω)$ when $Ω$ is countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of $Ω^Ω$ on a set $Ω$ of arbitrary infinite cardinality containing one of the following subgroups of $\sym(Ω)$: the pointwise stabilis…
A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $ω$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $ω$-semigroups, substantially generalizing the above mentioned …
This paper provides short proofs of two fundamental theorems of finite semigroup theory whose previous proofs were significantly longer, namely the two-sided Krohn-Rhodes decomposition theorem and Henckell's aperiodic pointlike theorem, using a new algebraic technique that we call the merge decomposition. A prototypical application of this technique decomposes a semigroup $T$ into a two-sided semidirect product whose components are built from two subsemigroups $T_1,T_2$, which together generate $T$, and the subsemigroup generated by their setwise product $T_1T_2$. In this sense we decompose $T…
Let $X$ be a finite or infinite chain and let $O(X)$ be the monoid of all endomorphisms of $X$. In this paper, we describe the largest regular subsemigroup of $O(X)$ and Green's relations on $O(X)$. In fact, more generally, if $Y$ is a nonempty subset of $X$ and $O(X,Y)$ the subsemigroup of $O(X)$ of all elements with range contained in $Y$, we characterize the largest regular subsemigroup of $O(X,Y)$ and Green's relations on $O(X,Y)$. Moreover, for finite chains, we determine when two semigroups of the type $O(X,Y)$ are isomorphic and calculate their ranks.
A description of all subsemigroups of $M_2(\mathbb{C})$ which are given by a countable intersection of constructible sets is given. Furthermore, it is shown that they are intersections of constructible semigroups.
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ? e2. Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempote…
The \emph{large rank} of a finite semigroup $Γ$, denoted by $r_5(Γ)$, is the least number $n$ such that every subset of $Γ$ with $n$ elements generates $Γ$. Howie and Ribeiro showed that $r_5(Γ) = |V| + 1$, where $V$ is a largest proper subsemigroup of $Γ$. This work considers the complementary concept of subsemigroups, called \emph{prime subsets}, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro's result about the large rank of Brandt semigroups. Further, this work obtains the large rank…
The multiplicative semigroup $M_n(F)$ of $n\times n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our gu…
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 build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of both the trace and inverse-kernel are discussed, in particular that both the trace and inverse-kernel maps are onto $\cap$-homomorphisms. The lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We use this to describe the lattice of left congruences on the b…
This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projec…
We prove that if $S$ is a $le$-semigroup in which left ideal elements commute (condition which is called $\mathbfΛ$), then any $\mathcal{J}$-class satisfying the Green condition is a subsemigroup of $S$. As a corollary of this we show that semisimple $le$-semigroups satisfying $\mathbfΛ$ are precisely those that decompose as a semilattice of left simple $\vee e$-semigroups which are in addition semisimple, intra-regular and satisfy $\mathbfΛ$.
Let $\nat^\nat$ be the semigroup of all mappings on the natural numbers $\nat$, and let $U$ and $V$ be subsets of $\nat^\nat$. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of $\nat^\nat$ such that $U$ is contained in the subsemigroup generated by $V$ and $C$. We give several results about the structure of the preorder $\preccurlyeq$. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder $\preccurlyeq$ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on $\nat$. The …
A semigroup is completely simple if it has no proper ideals and contains a primitive idempotent. We say that a completely simple semigroup $S$ is a homogeneous completely simple semigroup if any isomorphism between finitely generated completely simple subsemigroups of $S$ extends to an automorphism of $S$. Motivated by the study of homogeneous completely regular semigroups, we obtain a complete classification of homogeneous completely simple semigroups, modulo the group case. As a consequence, all finite regular homogeneous semigroups are described, thus extending the work of Cherlin on homoge…
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. If we insist on $a$ and $b$ being $\mathcal{R}$-related in $Q$, then we say that $S$ is a straight left I-order in $Q$. We give necessary and sufficient conditions for a semigroup to be a left I-order in a bisimple inverse $ω$-semigroup.
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.
