On Filters of Implicative Negatively Partially Ordered Ternary Semigroups

In this paper, we study a special set in an implicative n.p.o.(negatively partially ordered) ternary semigroup, and prove that a filter can be represented by the union of such sets. Indeed, let $(T, [\,\,\,],\leq,[\,\,\,]^*)$ be an implicative n.p.o. ternary semigroup. For any $a, b\in T$, we define $$S(a,b):=\{c\in T \,:\, [aa[bbc]^*]^*=1\}.$$ We have the following:\begin{enumerate} \item [(1)] A non-empty subset $F$ of $T$ isa filter if and only if it satisfies the following conditions: \begin{enumerate} \item[(F3)] $1\in F$; \item[(F4)] for any $a, b,c \in T$, if $[abc]^*\in F$ and $a,b \in F$, then $c \in F$. \end{enumerate} \item [(2)] If $T$ is commutative and $F$ is a filter of $T$, then $$F=\displaystyle\bigcup_{a,b\in F} S(a,b).$$\end{enumerate}