| Summary: | On a particular class of m-idempotent hyperrings, the relation <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>ξ</mi> <mi>m</mi> <mo>*</mo> </msubsup> </semantics> </math> </inline-formula> is the smallest strongly regular equivalence such that the related quotient ring is commutative. Thus, on such hyperrings, <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>ξ</mi> <mi>m</mi> <mo>*</mo> </msubsup> </semantics> </math> </inline-formula> is a new representation for the <inline-formula> <math display="inline"> <semantics> <msup> <mi>α</mi> <mo>*</mo> </msup> </semantics> </math> </inline-formula>-relation. In this paper, the <inline-formula> <math display="inline"> <semantics> <msub> <mi>ξ</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-parts on hyperrings are defined and compared with complete parts, <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-parts, and <i>m</i>-complete parts, as generalizations of complete parts in hyperrings. It is also shown how the <inline-formula> <math display="inline"> <semantics> <msub> <mi>ξ</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-parts help us to study the transitivity property of the <inline-formula> <math display="inline"> <semantics> <msub> <mi>ξ</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-relation. Finally, <inline-formula> <math display="inline"> <semantics> <msub> <mi>ξ</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-complete hyperrings are introduced and studied, stressing on the fact that they can be characterized by <inline-formula> <math display="inline"> <semantics> <msub> <mi>ξ</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>-parts. The symmetry plays a fundamental role in this study, since the protagonist is an equivalence relation, defined using also the symmetrical group of permutations of order <i>n</i>.
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