Natural partial order on semigroup partial linear transformation with restricted range

Abstract

The set with assosiative binary operations is semigroup. Of all set partial linear transformations with a composition function is semigroup which is called a partial linear transformation semigroup that is denoted by (PT (X), o). If Y is a subspace of X, with all images of PT (X) a subset of Y, so the semigroup which is formed is a semigroup of partial linear transformation with restricted range that is denoted by (PT(X,Y),o). A natural partial order is a relation which defined by a ≤ b if only if a = xb = by, xa = a for some x,y ∈PT(X,Y) in a semigroup partial linear transformation with restricted range. In this article using the literature study method, it discusses the characterization of natural partial order on partial linear transformations semigroups with restricted range resulting neccesary and sufficient condition for a natural partial order in semigroup partial linear transformation with restricted range.