On Filters of Implicative Negatively Partially Ordered Ternary Semigroups

Abstract

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 $T$ is
a 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}