On extensions of pseudo-valuations on BCK algebras

Busneag, Dumitru, Piciu, Dana and Istrata, Mihaela

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creative_2022_31_1_43_49

DOI: https://doi.org/10.37193/CMI.2022.01.04

Issue no: Vol 31/2022 no. 1

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In this paper we define a \textit{pseudo-valuation} on a BCK algebra $(A,\rightarrow ,1)$ as a real-valued function $v:A\rightarrow \mathbf{R}$ satisfying $v(1)=0$ and $v(x\rightarrow y)\geq v(y)-v(x),$ for every $x,y\in A$ $;$ $v$ is called a \textit{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 $d_{v}(x,y)=v(x\rightarrow y)+v(y\rightarrow x)$ for every $x,y\in A,$ where $\rightarrow$ 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.