Characterizations of ordered $h$-regular semirings by ordered $h$-ideals

Rukhshanda Anjum; Saad Ullah; Yu-Ming Chu; Mohammad Munir; Nasreen Kausar; Seifedine Kadry; Rukhshanda Anjum; Saad Ullah; Yu-Ming Chu; Mohammad Munir; Nasreen Kausar; Seifedine Kadry

doi:10.3934/math.2020370

AIMS Mathematics

2020, Volume 5, Issue 6: 5768-5790. doi: 10.3934/math.2020370

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Research article

Characterizations of ordered $h$-regular semirings by ordered $h$-ideals

1.
Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan
2.
Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
3.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China
4.
Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
5.
Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan
6.
Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Lebanon

Received: 24 March 2020 Accepted: 06 July 2020 Published: 10 July 2020
MSC : 16Y99, 16Y60

The objective of this paper is to study the ordered $h$-regular semirings by the properties of their ordered $h$-ideals. It is proved that each $h$-regular ordered semiring is an ordered $h$-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and $h$-regular semirings are given. It is also proved that each regular ordered semiring is an ordered $h$-regular semiring but the converse does not hold. The classifications of the left and the right ordered $h$-regular semirings and the left and the right ordered $h$-weakly regular semirings are also presented.
- semiring,
- ordered semiring,
- ordered $h$-regular,
- ordered $h$-ideal
Citation: Rukhshanda Anjum, Saad Ullah, Yu-Ming Chu, Mohammad Munir, Nasreen Kausar, Seifedine Kadry. Characterizations of ordered $h$-regular semirings by ordered $h$-ideals[J]. AIMS Mathematics, 2020, 5(6): 5768-5790. doi: 10.3934/math.2020370

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Abstract

The objective of this paper is to study the ordered $h$-regular semirings by the properties of their ordered $h$-ideals. It is proved that each $h$-regular ordered semiring is an ordered $h$-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and $h$-regular semirings are given. It is also proved that each regular ordered semiring is an ordered $h$-regular semiring but the converse does not hold. The classifications of the left and the right ordered $h$-regular semirings and the left and the right ordered $h$-weakly regular semirings are also presented.

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