On $ n $-ary ring congruences of $ n $-ary semirings

Pakorn Palakawong na Ayutthaya; Bundit Pibaljommee; Pakorn Palakawong na Ayutthaya; Bundit Pibaljommee

doi:10.3934/math.20221019

AIMS Mathematics

2022, Volume 7, Issue 10: 18553-18564. doi: 10.3934/math.20221019

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Research article

On $ n $-ary ring congruences of $ n $-ary semirings

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 30 June 2022 Revised: 09 August 2022 Accepted: 14 August 2022 Published: 18 August 2022
MSC : 06F25, 16Y60

In universal algebra, it is well-known that if $ S $ is an algebraic structure, then the kind of algebraic structure of $ S/\rho $ is similar to $ S $ where $ \rho $ is a congruence relation on $ S $. In this work, we study the notion of a full $ k $-ideal $ A $ of an $ n $-ary semiring $ S $ and construct a congruence relation $ \rho $ on $ S $ with respect to the full $ k $-ideal $ A $ in order to make the quotient $ n $-ary semiring $ S/\rho $ to be an $ n $-ary ring. Moreover, the notion of an $ h $-ideal of an $ n $-ary semiring was studied and connections between an $ h $-ideal and a $ k $-ideal of an $ n $-ary semiring were investigated.
- $ n $-ary ring,
- $ n $-ary semiring,
- ring congruence,
- $ k $-ideal,
- $ h $-ideal
Citation: Pakorn Palakawong na Ayutthaya, Bundit Pibaljommee. On $ n $-ary ring congruences of $ n $-ary semirings[J]. AIMS Mathematics, 2022, 7(10): 18553-18564. doi: 10.3934/math.20221019

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Abstract

In universal algebra, it is well-known that if $ S $ is an algebraic structure, then the kind of algebraic structure of $ S/\rho $ is similar to $ S $ where $ \rho $ is a congruence relation on $ S $. In this work, we study the notion of a full $ k $-ideal $ A $ of an $ n $-ary semiring $ S $ and construct a congruence relation $ \rho $ on $ S $ with respect to the full $ k $-ideal $ A $ in order to make the quotient $ n $-ary semiring $ S/\rho $ to be an $ n $-ary ring. Moreover, the notion of an $ h $-ideal of an $ n $-ary semiring was studied and connections between an $ h $-ideal and a $ k $-ideal of an $ n $-ary semiring were investigated.

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