Valuation on L-Algebras

Rajab Ali Borzooei; Mona Aaly Kologani; Mohammad Mohseni Takallo; Xiao Long Xin

doi:10.18778/0138-0680.2026.20

Authors

Rajab Ali Borzooei Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Author https://orcid.org/0000-0001-7538-7885
Mona Aaly Kologani Department of Mathematics, Hatef Higher Education Institute, Zahedan, Iran Author https://orcid.org/0000-0002-5234-2876
Mohammad Mohseni Takallo School of Science, Xi’an Polytechnic University, Xi’an, China Author https://orcid.org/0000-0002-4113-3657
Xiao Long Xin School of Bussiness, Xi’an International University, Xi’an, China Author https://orcid.org/0000-0002-8495-7322

DOI:

https://doi.org/10.18778/0138-0680.2026.20

Keywords:

L-algebra, valuation map, 1-valuation map, topology, metric space

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.

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