For a non-empty set X denote the full transformation semigroup on X by T(X) and suppose that \(\sigma \) is an equivalence relation on X. For every \(f\in T(X)\), the kernel of f is defined to be \(\ker f =\{(x, y)\in X\times X\mid f(x) = f(y)\}\). Evidently, \(E(X, \sigma )=\{f\in T(X) \mid \sigma \subseteq \ker f\}\) is a subsemigroup of T(X). Also, the subset \(RE(X, \sigma )\) of \(E(X, \sigma )\) consisting of regular elements is a subsemigroup. Partition of a semigroup by Green’s \(*\)-relations was first introduced by Fountain in 1979 and the Green’s \(\sim \)-relations (with respect to a non-empty subset U of the set of idempotents) as a new method of partition were introduced by Lawson (J Algebra 141(2):422–462, 1991). In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup \(E(X, \sigma )\). This investigation proves that the semigroup \(E(X, \sigma )\) is always a right Ehresmann semigroup. Finally, we prove that \(RE(X, \sigma )\) is an orthodox semigroup if and only if the set X consists of at most two \(\sigma \)-classes.

对于非空集X分别表示在全变换半群X由Ť（X），并假设\（\西格玛\）是一个等价关系X。对于每个\ [f \ in T（X）\），f的内核定义为\（\ ker f = \ {（x，y）\ in X \ times X \ mid f（x）= f（ y）\} \）。显然，\（E（X，\ sigma）= \ {f \ in T（X）\ mid \ sigma \ subseteq \ ker f \} \）是T（X）的一个子半群。此外，该子集\（RE（X，\西格马）\）的\（E（X，\西格马）\）由常规元素组成的是一个亚半群。1979年，Fountain首次引入了由Green的\（* \）-关系对半群进行划分，而Green的\（\ sim \）-关系（相对于幂等集的非空子集U ）首次引入。分区方法由Lawson（J Algebra 141（2）：422-462，1991）引入。在本文中，我们打算对半群\（E（X，\ sigma）\）的这两套Green关系进行某些刻画。该研究证明，半群\（E（X，\ sigma）\）始终是正确的Ehresmann半群。最后，我们证明\（RE（X，\ sigma）\）是正统半群，当且仅当集合X最多包含两个\（\ sigma \）-类。