Some subvarieties of semiring variety COS$ ^{+}_{3} $

Xuliang Xian; Yong Shao; Junling Wang; Xuliang Xian; Yong Shao; Junling Wang

doi:10.3934/math.2022237

AIMS Mathematics

2022, Volume 7, Issue 3: 4293-4303. doi: 10.3934/math.2022237

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Some subvarieties of semiring variety COS$ ^{+}_{3} $

School of Mathematics, Northwest University, Xi'an 710127, Shaanxi, China

Received: 12 July 2021 Revised: 26 November 2021 Accepted: 06 December 2021 Published: 20 December 2021
MSC : 03C05, 08A30, 17A99

In this paper, we study some subvarieties of a semiring variety determined by certain additional identities. We first present alternative characterizations for equivalences $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{D}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{D}} $. Then we give the sufficient and necessary conditions for these equivalences to be congruence. Finally, we prove that semiring classes defined by these congruences are varieties and provide equational bases.
- semiring,
- Green's relation,
- congruence,
- variety of semirings,
- subvariety
Citation: Xuliang Xian, Yong Shao, Junling Wang. Some subvarieties of semiring variety COS$ ^{+}_{3} $[J]. AIMS Mathematics, 2022, 7(3): 4293-4303. doi: 10.3934/math.2022237

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Abstract

In this paper, we study some subvarieties of a semiring variety determined by certain additional identities. We first present alternative characterizations for equivalences $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{D}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{D}} $. Then we give the sufficient and necessary conditions for these equivalences to be congruence. Finally, we prove that semiring classes defined by these congruences are varieties and provide equational bases.

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