Admitting a Semihyperring with Zero of Certain Linear Transformation Subsemigroups of $L_R(V,W)$ (Part II)
บทคัดย่อ
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 2 A
(called a zero) such that x+0 = 0+x = fxg and x¢0 = 0¢x = 0 for all x 2 A. For
a semigroup S, let S0 be S if S has a zero and S contains more than one element,
otherwise, let S0 be the semigroup S with a zero adjoined. We say that a semi-
group S is said to admit a semihyperring with zero if there exists a hyperoperation
+ on S0 such that (S0;+; ¢) is a semihyperring with zero 0 where ¢ is the operation
on S0 and 0 is the zero of S0. Let V be a vector space over a division ring R, W a
subspace of V and LR(V;W) the semigroup under composition of all linear trans-
formations from V into W. For each ® 2 LR(V;W), let F(®) consist of all elements
in V ¯xed by ®. Denote by OMR(V;W), OER(V;W), AIR(V ;W) and AIR(V;W)
the set of all linear transformations ® in LR(V;W) where dimR Ker ® are in¯nite,
the set of all linear transformations ® in LR(V;W) where dimR(W=Im®) are in¯-
nite, the set of all linear transformations ® in LR(V;W) where dimR(V=F(®)) are
¯nite and the set of all linear transformations ® in LR(V;W) where dimR(W=F(®))
are ¯nite, respectively. Moreover, let H and S be subsemigroups of AIR(V ;W)
and AIR(V;W), respectively.
We show that OMR(V;W) [ H, OER(V;W) [ H, OMR(V;W) [ S and
OER(V;W)[S are semigroups. Furthermore, we determine whether or when they
admit the structure of a semihyperring with zero.