Let V be a linear space over a field F with finite dimension,L(V) the semigroup,under composition,of all linear transformations from V into itself.Suppose that V = V1⊕V2⊕···⊕Vm is a direct sum decomp...Let V be a linear space over a field F with finite dimension,L(V) the semigroup,under composition,of all linear transformations from V into itself.Suppose that V = V1⊕V2⊕···⊕Vm is a direct sum decomposition of V,where V1,V2,...,Vm are subspaces of V with the same dimension.A linear transformation f ∈ L(V) is said to be sum-preserving,if for each i(1 ≤ i ≤ m),there exists some j(1 ≤ j ≤ m) such that f(Vi) ■Vj.It is easy to verify that all sum-preserving linear transformations form a subsemigroup of L(V) which is denoted by L⊕(V).In this paper,we first describe Green's relations on the semigroup L⊕(V).Then we consider the regularity of elements and give a condition for an element in L⊕(V) to be regular.Finally,Green's equivalences for regular elements are also characterized.展开更多
文摘Let V be a linear space over a field F with finite dimension,L(V) the semigroup,under composition,of all linear transformations from V into itself.Suppose that V = V1⊕V2⊕···⊕Vm is a direct sum decomposition of V,where V1,V2,...,Vm are subspaces of V with the same dimension.A linear transformation f ∈ L(V) is said to be sum-preserving,if for each i(1 ≤ i ≤ m),there exists some j(1 ≤ j ≤ m) such that f(Vi) ■Vj.It is easy to verify that all sum-preserving linear transformations form a subsemigroup of L(V) which is denoted by L⊕(V).In this paper,we first describe Green's relations on the semigroup L⊕(V).Then we consider the regularity of elements and give a condition for an element in L⊕(V) to be regular.Finally,Green's equivalences for regular elements are also characterized.
