<abstract xmlns="http://www.w3.org/1999/xhtml">The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure <italic>R/</italic>α* is a commutative ring. In this paper we introduce on hyperrings the relation ζ<italic>m</italic>, which is smaller than α*, and show that, on a particular class of <italic>m</italic>-idempotent hyperrings <italic>R</italic>, it is the smallest strongly regular relation such that the quotient ring <italic>R/ζ</italic>*<italic>m</italic> is commutative. Some properties of this new relation and its differences from the α* -relation are illustrated and discussed. Finally, we show that ζ<italic>m</italic> is a new representation for α* on this particular class of <italic>m</italic>-idempotent hyperrings.</abstract>

The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure R/α* is a commutative ring. In this paper we introduce on hyperrings the relation ζm, which is smaller than α*, and show that, on a particular class of m-idempotent hyperrings R, it is the smallest strongly regular relation such that the quotient ring R/ζ*m is commutative. Some properties of this new relation and its differences from the α* -relation are illustrated and discussed. Finally, we show that ζm is a new representation for α* on this particular class of m-idempotent hyperrings.

The commutative quotient structure of <italic>m</italic>-idempotent hyperrings

Department of Mathematics, Faculty of Basic Sciences

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The commutative quotient structure of -idempotent hyperrings

The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure R/α* is a...

The commutative quotient structure of m-idempotent hyperrings

Published Online: Apr 09, 2020

Page range: 219 - 236

Received: Apr 23, 2019

Accepted: May 31, 2019

DOI: https://doi.org/10.2478/auom-2020-0015

Keywords
Hyperring, -idempotent hyperring, strongly regular relation

© 2020 Azam Adineh Zadeh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The commutative quotient structure of m-idempotent hyperrings

Published Online: Apr 09, 2020

Page range: 219 - 236

Received: Apr 23, 2019

Accepted: May 31, 2019

DOI: https://doi.org/10.2478/auom-2020-0015

KeywordsHyperring, -idempotent hyperring, strongly regular relation

© 2020 Azam Adineh Zadeh et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Hyperring, -idempotent hyperring, strongly regular relation