(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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, [ ],≤,[ ]^*) be an implicative n.p.o. ternary semigroup. For any a, bT, we define S(a,b):={cT : [aa[bbc]^*]^*=1}. We have the following: enumerate [(1)] A non-empty subset F of T is a filter if and only if it satisfies the following conditions: enumerate [(F3)] 1F; [(F4)] for any a, b,c T, if [abc]^*F and a,b F, then c F. enumerate [(2)] If T is commutative and F is a filter of T, then F=_a,bF S(a,b). enumerate