On extensions of pseudo-valuations on BCK algebras

Abstract

"In this paper we define a pseudo-valuation on a BCK algebra (A,→, 1) as a real-valued function v : A → R satisfying v(1) = 0 and v(x → y) ≥ v(y) − v(x), for every x, y ∈ A ; v is called a valuation if x = 1 whenever v(x) = 0. We prove that every pseudo-valuation (valuation) v induces a pseudo-metric (metric) on A defined by dv(x, y) = v(x → y) + v(y → x) for every x, y ∈ A, where → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations) on BCK algebras. In this paper we define a pseudo-valuation on a BCK algebra (A,→, 1) as a real-valued function v : A → R satisfying v(1) = 0 and v(x → y) ≥ v(y) − v(x), for every x, y ∈ A ; v is called a valuation if x = 1 whenever v(x) = 0. We prove that every pseudo-valuation (valuation) v induces a pseudo-metric (metric) on A defined by dv(x, y) = v(x → y) + v(y → x) for every x, y ∈ A, where → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations) on BCK algebras. In this paper we define a pseudo-valuation on a BCK algebra (A,→, 1) as a real-valued function v : A → R satisfying v(1) = 0 and v(x → y) ≥ v(y) − v(x), for every x, y ∈ A ; v is called a valuation if x = 1 whenever v(x) = 0. We prove that every pseudo-valuation (valuation) v induces a pseudo-metric (metric) on A defined by dv(x, y) = v(x → y) + v(y → x) for every x, y ∈ A, where → is uniformly continuous in both variables. The aim of this paper is to provide several theorems on extensions of pseudo-valuations (valuations) on BCK algebras."