Valuation on L-Algebras

Abstract

The purpose of this paper is to define the notion of valuation map on \(L\)-algebras and introduce its properties. For this, the concepts of valuation and 1-valuation maps are introduced and some characterizations and related properties are investigated. Using a valuation map, a congruence relation is obtained and proved that this congruence relation is equivalent to the congruence relation made by an ideal. Also, showed that the image and inverse image of any valuation is a valuation. Finally, by using a 1-valuation map \(\xi\) on \(L\)-algebra \(\mathcal{L}\), an \(L\)-algebra \((\widetilde{\mathcal{L}},\ast,\widetilde{\{1\}})\) and a metric space \((\widetilde{\mathcal{L}}, \widetilde{d_\xi})\) has been obtained.