Dedekind $ \sigma $-complete partially ordered rings

Jingjing Ma; Jingjing Ma

doi:10.3934/era.2026164

Electronic Research Archive

2026, Volume 34, Issue 6: 3643-3654. doi: 10.3934/era.2026164

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Dedekind $ \sigma $-complete partially ordered rings

Jingjing Ma ^,

Department of Mathematical, Applied and Physical Sciences, University of Houston-Clear Lake, 2700 Bay Area Blvd., Houston TX 77058, USA

Received: 07 March 2026 Revised: 13 April 2026 Accepted: 20 April 2026 Published: 29 April 2026

In this paper, I present the properties and structures of Dedekind $ \sigma $-complete directed partially ordered rings and algebras over $ \mathbb{R} $. Under certain conditions, such as property P₁, chain conditions on convex ideals, and strong Archimedean, structures are isomorphic to the real number field $ \mathbb{R} $ with the usual total order or finite product or subdirect product of $ \mathbb{R} $.
- Archimedean,
- Dedekind $ \sigma $-complete,
- idempotent,
- infinite prime,
- maximal partial order,
- strong Archimedean,
- total order
Citation: Jingjing Ma. Dedekind $ \sigma $-complete partially ordered rings[J]. Electronic Research Archive, 2026, 34(6): 3643-3654. doi: 10.3934/era.2026164

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Abstract

In this paper, I present the properties and structures of Dedekind $ \sigma $-complete directed partially ordered rings and algebras over $ \mathbb{R} $. Under certain conditions, such as property P₁, chain conditions on convex ideals, and strong Archimedean, structures are isomorphic to the real number field $ \mathbb{R} $ with the usual total order or finite product or subdirect product of $ \mathbb{R} $.

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