In this note, some conditions are investigated under which the left amenability of a semigroup S is a consequence of the left amenability of its subsemigroups. It is known that for the Green’s relation $\mathcal {H}^S$ on S, an $\mathcal {H}^S$-class of S is a semigroup if and only if it is a subgroup of S, and hence it contains a unique identity. Let S be a semigroup such that every $\mathcal {H}^S$-class of S is a group and E, the set of idempotents of S, is a subsemigroup of S. As the main result of this note, applying the above fact, a connection between left amenability of S, left amenability of E, and left amenability of its $\mathcal {H}^S$-classes is established.

As an application, I completely determine left amenable Clifford semigroups and left amenable rectangular groups, when they are left amenable with some measure such that the union of every collection of $\mathcal {H}^S$-classes of S with zero measure has zero measure (especially, when E is finite or when E is countable and it is left amenable with a measure which is countably additive). Indeed, I show that under this assumption, (i) a Clifford semigroup S is left amenable if and only if E has a zero element z and $H_z$, the $\mathcal {H}^S$-class of S which contains z, is a left amenable group and (ii) a rectangular group S is left amenable if and only if it is a right group and its $\mathcal {H}^S$-classes are left amenable groups.

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$, we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $, and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $. We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$. This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

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.

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 1975, Symons described the automorphisms of the semigroup T(X,Y ) consisting of all total transformations from a set X into a fixed subset Y of X. Recently Sanwong, Singha and Sullivan determined all maximal (and all minimal) congruences on T(X,Y ), and Sommanee studied Green’s relations in T(X,Y ). Here, we describe Green’s relations and ideals for the semigroup T(V,W) consisting of all linear transformations from a vector space V into a fixed subspace W of V.