Suppose σ is an equivalence on a set X and let E(X, σ) denote the semigroup (under composition) of all α: X → X such that σ ⊆ α ∘ α −1. Here we characterise Green’s relations and ideals in E(X, σ). This is analogous to recent work by Sullivan on K(V, W), the semigroup (under composition) of all linear transformations β of a vector space V such that W ⊆ ker β, where W is a fixed subspace of V.

If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). If θ ∈ P(Y, X), then P(X, Y) is a so-called "generalized semigroup" of transformations under the "sandwich operation": α * β = α ◦ θ ◦ β, for each α, β ∈ P(X, Y). We denote this semigroup by P(X, Y, θ) and, in this paper, we characterize Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(X, Y, θ) are equal. This solves a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(X, Y, θ) and characterize when these semigroups satisfy certain regularity conditions.

Let Y be a fixed nonempty subset of a set X and let T (X, Y ) denote the semigroup of all total transformations from X into Y . In 1975, Symons described the automorphisms of T (X, Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T (X, Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T (X, Y ) when |Y | = 1 and Y = X ; and using this, they described the Green's relations on T (X, Y ). Here, we use their work to describe the ideal structure of T (X, Y ). We also correct the proof of the corresponding result for a linear analogue of T (X, Y ).2010 Mathematics subject classification: primary 20M20; secondary 15A04.

Yang (1999) classified the maximal inverse subsemigroups of all the ideals of the symmetric inverse semigroup I X defined on a finite set X. Here we do the same for the semigroup I V of all one-to-one partial linear transformations of a finitedimensional vector space. We also show that I X is almost never isomorphic to I V for any set X and any vector space V , and prove that any inverse semigroup can be embedded in some I V .

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