On pseudo-valuations on BCK-algebras

Abstract

In this paper, we study some properties of pseudo-valuations and their induced quasi metrics. The continuity of operation of a BCK-algebra was studied with topology induced by a pseudo-valuation. Moreover, we show that product of finite number of this pseudo metric spaces is a pseudo metric space. Also, we prove that if a BCK-algebra X has a pseudo-valuation, then every quotient space of X has a pseudo metric. The completion of this spaces has been investigated in the present study. 2. Preliminaries 2.1. BCK-algebras An algebra (X, * , 0) of type (2, 0) is called a BCK-algebra if it satisfies the following axioms: for any x, y, z ∈ X, (1) ((x * y) * (x * z)) * (z * y) = 0, (2) (x * (x * y)) * y = 0,

On extensions of pseudo-valuations on BCK algebras

“…In [1], Busneag defined pseudo-valuations on Hilbert algebras and proved that every pseudo-valuation induced a pseudo-metric. Using this model, in [4], [5], [7], [11], [12], [14], [15], [16] is introduced the notion of pseudo-valuation on BCK, BCI, BCC algebras and several properties are discussed.…”

Pseudo-Valuations on UP-Algebras

“…Jun, Lee and Song analyzed in article [8] several types of quasi-valuation maps on BCK -algebra and their interactions. Also, Mehrshad and Kouhestani were interested in pseudo-valuations on BCK -algebra [10]. Jun, Ahn and Roh.…”

Untitled

“…Based on the concept of pseudo-valuation in R 0 -algebras, Yang and Xin [7] characterized pseudo pre-valuations on EQ-algebras. Mehrshad and Kouhestani studied Pseudo-Valuations on BCK-Algebras [8]. Pseudo-valuations on a BCC-algebra was introduced by Jun et al [9], where they have shown that binary operation in BCC-algebras is uniformly continuous.…”