Linear transformation subsemigroups of LR(V,W) admitting the structure of a semihyperring with zero

Abstract

A semihyperring with zero is a triple (A,+,#) such that (A,+) is a semihypergroup, (A,#) is a semigroup, # is distributive over + and there exists 0(called a zero) in A such that x+0=0+x is singleton x and x#0=0#x=0 for all x in A. We say that a semigroup S admits the structure of a semihyperring with zero if there exists a hyperoperation + on S with zero such that (S with zero,+,#) is a semihyperring with zero where # is the operation on S. Let V be a vector space over a division ring R , W a subspace of V and LR(V,W) the semigroup of all linear transformations from V into W under composition. For each q in LR(V,W) , let F(q) consist of all elements in V fixed by q. Let OMR(V,W) be set of all linear transformations from V into W such that dimension of kernel is infinite, OER(V,W) be set of q in LR(V,W) such that dimension of (W/Im q) is infinite, GR(V,W) be set of q in LR(V,W) such that q restrict on W is isomorphism, AIR(V,W*) be set of q in LR(V,W) such that dimension of (W/F(q)) is finite and AIR(V*,W) be set of q in LR(V,W) such that dimension of(V/F(q)) is finite. Moreover, let H, S and T be subsemigroups of GR(V,W), AIR(V,W*) and AIR(V*,W), respectively. We show that OMR(V,W) and OER(V,W) union H, S and T are semigroups. Furthermore, we determine whether they admit the structure of a semihyperring with zero.