Metrization of idempotent semiring of topological rings

Sunil Kumar Maity

https://orcid.org/0000-0002-3155-6162

India

University of Calcutta

Department of Pure mathematics

Gopal Adak

https://orcid.org/0009-0005-7330-3680

India

University of Calcutta

Department of Pure mathematics

Submitted: 2024-11-20

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Accepted: 2025-03-06

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Published: 2025-10-01

DOI: https://doi.org/10.4995/agt.2025.22854

Copyright (c) 2025 Sunil Kumar Maity, Gopal Adak

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Keywords:

$\mathscr{H}^ $-pseudo-valuation, $\mathscr{H}^ $-norm, topological completely regular (algebraic) semiring, b-lattice, uniformity

Supporting agencies:

This research was not funded

Abstract:

In this paper, we have introduced natural H+ - pseudovaluation and  H+ -norm on a completely regular semiring making it a topological completely regular (algebraic) semiring and established some necessary and sufficient conditions of metrizability of a topological completely regular (algebraic) semiring. We have generalized the Birkhoff-Kakutani theorem on a first countable topological completely regular (algebraic) semiring. Also we have established that a topological completely regular (algebraic) semiring is metrizable if and only if it is an M- space and its set of additive idempotents is a metrizable Gδ - set in S. Finally we have studied the completion of a b- lattice of topological rings in its uniformity structure.

 

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