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 implicativehomomorphism is a homomorphism. Let φ : 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/ρKer φ, where ρKer φ is a congruence relation defined by Ker φ. We prove that there exists an implicative homomorphism ψ such that ψ ◦ η = φ, where η is a canonical homomorphism from T1 onto T1/ρKerφ.