On Po-injective and Po-surjective Wreath Product of Pomonoids

Abstract

Let $R$  and $S$ be pomonoids and $_{R}{A}$ be a left $R$-poset. The wreath product of the pomonoids $R$  and $S$ by $_{R}{A}$  is defined as the pomonoid $T~=~R \times F(A, S)$ While, the wreath product $_TC$ of the left $R$-poset $_{R}{A}$ with the left $S$-poset $_{S}{B}$ over the pomonoid $T= R \times F(A, S)$ is the left $T$-poset ${_T C}= {_R A}\times {_S B}$ endowed with the monotone action given by  $(r, f)(a, b) = (ra, f(a)b),$ where  $(r, f) \in R\times F(A, S)$ and $ (a, b) \in A\times B$. The po-injectivity and po-cancellative properties on the wreath product $_TC$ are studied and the relations between them are established. The relation between po-surjective property and other properties on the wreath product $_TC$ are also established. Finally the characterization of some properties of po-flatness such as po-torsion free, properties $(P)$, $(E)$, $(P_E)$, and  strongly flat have been examined on  the wreath product $_TC$ and the relations among them have  also been established.