We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that

$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$

$T(V,W)$

$T(V,W)$

$Q$

$W$

$V$

$Q$

$W$

$n$

$m$

$V$

$F$

We calculate the rank and idempotent rank of the semigroup ${\mathcal{E}}(X,{\mathcal{P}})$ generated by the idempotents of the semigroup ${\mathcal{T}}(X,{\mathcal{P}})$ which consists of all transformations of the finite set $X$ preserving a nonuniform partition ${\mathcal{P}}$. We also classify and enumerate the idempotent generating sets of minimal possible size. This extends results of the first two authors in the uniform case.

In this short note, we describe all the elements in the semigroup

$$\begin{eqnarray}S(X,Y)=\{f\in {\mathcal{T}}_{X}:f(Y)\subseteq Y\}\end{eqnarray}$$

Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

For a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote

$$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$

${T}_{\exists } (X)$

${ \mathcal{T} }_{X} $

${T}_{\exists } (X)$

For an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$. For $ \alpha \in \mathcal {I}(X)$, let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$. For an arbitrary $ \alpha \in \mathcal {I}(X)$, we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$, describe the regular elements of $C(\alpha )$, and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$, we determine the structure of $C(\alpha )$in terms of Green’s relations.

Let 𝒯X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

We devise a fairly general sufficient condition ensuring that the endomorphism monoid of a countably infinite ultrahomogeneous structure (i.e. a Fraïssé limit) embeds all countable semigroups. This approach not only provides us with a framework unifying the previous scattered results in this vein, but actually yields new applications for endomorphism monoids of the (rational) Urysohn space and the countable universal ultrahomogeneous semilattice.

Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y )={α∈P(X):Xα⊆Y } and defined I(X,Y ) to be the set of all injective transformations in PT(X,Y ) . Hence PT(X,Y ) and I(X,Y ) are subsemigroups of P(X) . In this paper, we study properties of the so-called natural partial order ≤ on PT(X,Y ) and I(X,Y ) in terms of domains, images and kernels, compare ≤ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y ) and I(X,Y ) which are compatible. Also, the minimal and maximal elements are described.

We show that if A is a stable basis algebra satisfying the distributivity condition, then B is a reduct of an independence algebra A having the same rank. If this rank is finite, then the endomorphism monoid of B is a left order in the endomorphism monoid of A.

For an infinite set X, denote by Γ(X) the semigroup of all injective mappings from X to X. For α∈Γ(X), let C(α)={β∈Γ(X):αβ=βα} be the centralizer of α in Γ(X). For an arbitrary α∈Γ(X), we characterize the elements of C(α) and determine Green’s relations in C(α), including the partial orders of ℒ-, ℛ-, and 𝒥-classes.

Suppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup . For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.

In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

If $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

For a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

In 1986, Higgins proved that T(X), the semigroup (under composition) of all total transformations of a set X, has a proper dense subsemigroup if and only if X is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup T(X, Y) consisting of all total transformations from X into Y under the operation α * β = αθβ, where θ is any fixed element of T(Y, X). We show that this semigroup has a proper dense subsemigroup if and only if X and Y are infinite and | Yθ| = min{|X|,|Y|}, and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.