Semilattice strongly regular relations on ordered $ n $-ary semihypergroups

Jukkrit Daengsaen; Sorasak Leeratanavalee; Jukkrit Daengsaen; Sorasak Leeratanavalee

doi:10.3934/math.2022031

AIMS Mathematics

2022, Volume 7, Issue 1: 478-498. doi: 10.3934/math.2022031

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Semilattice strongly regular relations on ordered $ n $-ary semihypergroups

Jukkrit Daengsaen ¹,
Sorasak Leeratanavalee ^{2
,
,}

1.
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2.
Research Group in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received: 25 April 2021 Accepted: 30 August 2021 Published: 13 October 2021
MSC : 20N20, 20N15, 06F05

In this paper, we introduce the concept of $ j $-hyperfilters, for all positive integers $ 1\leq j \leq n $ and $ n \geq 2 $, on (ordered) $ n $-ary semihypergroups and establish the relationships between $ j $-hyperfilters and completely prime $ j $-hyperideals of (ordered) $ n $-ary semihypergroups. Moreover, we investigate the properties of the relation $ \mathcal{N} $, which is generated by the same principal hyperfilters, on (ordered) $ n $-ary semihypergroups. As we have known from ^[21] that the relation $ \mathcal{N} $ is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on $ n $-ary semihypergroups where $ n\geq 3 $. However, we provide a sufficient condition that makes the previous conclusion true on $ n $-ary semihypergroups and ordered $ n $-ary semihypergroups where $ n\geq 3 $. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their $ \mathcal{N} $-classes. As an application of the results, a related problem posed by Tang and Davvaz in ^[31] is solved.
- ordered semihypergroup,
- $ n $-ary semihypergroup,
- hyperideal,
- hyperfilter
Citation: Jukkrit Daengsaen, Sorasak Leeratanavalee. Semilattice strongly regular relations on ordered $ n $-ary semihypergroups[J]. AIMS Mathematics, 2022, 7(1): 478-498. doi: 10.3934/math.2022031

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Abstract

In this paper, we introduce the concept of $ j $-hyperfilters, for all positive integers $ 1\leq j \leq n $ and $ n \geq 2 $, on (ordered) $ n $-ary semihypergroups and establish the relationships between $ j $-hyperfilters and completely prime $ j $-hyperideals of (ordered) $ n $-ary semihypergroups. Moreover, we investigate the properties of the relation $ \mathcal{N} $, which is generated by the same principal hyperfilters, on (ordered) $ n $-ary semihypergroups. As we have known from ^[21] that the relation $ \mathcal{N} $ is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on $ n $-ary semihypergroups where $ n\geq 3 $. However, we provide a sufficient condition that makes the previous conclusion true on $ n $-ary semihypergroups and ordered $ n $-ary semihypergroups where $ n\geq 3 $. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their $ \mathcal{N} $-classes. As an application of the results, a related problem posed by Tang and Davvaz in ^[31] is solved.

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