Double-framed soft <em>h</em>-semisimple hemirings

Faiz Muhammad Khan; Weiwei Zhang; Hidayat Ullah Khan; Faiz Muhammad Khan; Weiwei Zhang; Hidayat Ullah Khan

doi:10.3934/math.2020438

AIMS Mathematics

2020, Volume 5, Issue 6: 6817-6840. doi: 10.3934/math.2020438

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Research article

Double-framed soft h-semisimple hemirings

1.
Department of Mathematics and Statistics, University of Swat, KP, Pakistan
2.
School of Mathematics and Statistics, Northwestern Polytechnical University Xi’an, Shaanxi, P. R. China
3.
Department of Mathematics, University of Malakand, KP, Pakistan

Received: 24 July 2020 Accepted: 21 August 2020 Published: 03 September 2020
MSC : 08Axx, 08A72, 16Wxx

In dearth of parameterization, various uncertain-ordinary theories like the theory of fuzzy sets and the theory of probability, which failed to address the emergence of modern day sophisticated, complex, and unpredictable problems of various disciplines such as economics and engineering. We aim to provide an appropriate mathematical tool for resolving such complicated problems with the initiation and conceptualization of the notion of double-framed soft sets in hemirings. In the structural theory, h-ideals of hemirings play a key role, therefore, new types of h-ideals of a hemiring R known as double-framed soft h-interior ideals and double-framed soft h-ideals are determined. It is shown that every double-framed soft h-ideal of R is double-framed soft h-interior ideal but the converse is not true which is verified through suitable examples. Further, the conditions under which both these concepts coincide are provided. More precisely, if a hemiring R is h-hemiregular, (resp. h-intra-hemiregular, h-semisimple), then every double-framed soft h-interior ideal of R will be double-framed soft h-ideal. Several classes of hemirings such as h-hemiregular, h-intra-hemiregular, h-simple and h-semisimple are characterized by the notion of double-framed soft h-interior ideals.
- DFS sets,
- DFS h-ideal,
- DFS h-interior ideal,
- h-hemiregular hemiring,
- h-simple hemiring,
- h-semisimple hemiring
Citation: Faiz Muhammad Khan, Weiwei Zhang, Hidayat Ullah Khan. Double-framed soft h-semisimple hemirings[J]. AIMS Mathematics, 2020, 5(6): 6817-6840. doi: 10.3934/math.2020438

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Abstract

In dearth of parameterization, various uncertain-ordinary theories like the theory of fuzzy sets and the theory of probability, which failed to address the emergence of modern day sophisticated, complex, and unpredictable problems of various disciplines such as economics and engineering. We aim to provide an appropriate mathematical tool for resolving such complicated problems with the initiation and conceptualization of the notion of double-framed soft sets in hemirings. In the structural theory, h-ideals of hemirings play a key role, therefore, new types of h-ideals of a hemiring R known as double-framed soft h-interior ideals and double-framed soft h-ideals are determined. It is shown that every double-framed soft h-ideal of R is double-framed soft h-interior ideal but the converse is not true which is verified through suitable examples. Further, the conditions under which both these concepts coincide are provided. More precisely, if a hemiring R is h-hemiregular, (resp. h-intra-hemiregular, h-semisimple), then every double-framed soft h-interior ideal of R will be double-framed soft h-ideal. Several classes of hemirings such as h-hemiregular, h-intra-hemiregular, h-simple and h-semisimple are characterized by the notion of double-framed soft h-interior ideals.

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