Implicative Negatively Partially Ordered Ternary Semigroups

In this paper, we introduce and examine the notion of implicative negatively partially ordered ternary semigroups, for short implicative n.p.o. ternary semigroup, which include an element that serves as both the greatest element and the multiplicative identity. We study the notion of implicative homomorphisms between these ternary semigroups, and have that any implicative homomorphism is a homomorphism. Let phi : T1 -> T2 be an implicative homomorphism from a commutative implicative n.p.o. ternary semigroup T1 onto T2. We construct a quotient commutative implicative n.p.o. ternary semigroup T1/rho Ker phi, where rho Ker phi is a congruence relation defined by Ker phi. We prove that there exists an implicative homomorphism psi such that psi degrees eta = phi, where eta is a canonical homomorphism from T1 onto T1/rho Ker phi.