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The Relation % and Minimal bi-ideals in r-semigroups

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2014

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Author Name
Abstract (2. Language): 
In this paper we introduce the relation % "to generate the same principal bi-ideal" in r-semigroups. One of the main results that are proved here is the analogue of the Green's Theorem for r-semigroups, which we call the Green's Theorem for the relation % in r-semigroups. Applying our Green's Theorem for relation % in r-semigroups, we prove that any bi-ideal of a r-semigroup without zero is minimal if and only if it is a r-subgroup. Further, we prove that, if a r-semigroup M without zero has a cancellable element contained in a minimal bi-ideal B of M, then M is a r-group. Finally, we prove that, if for elements a, c of a r-semigroup without zero we have atyc and the principal bi-ideal (a)b and principal quasi-ideal (a)q are minimal, then (a)b = (a)q and the principal bi-ideal (c)b and the principal quasi-ideal (c)q are minimal too, and (c)b = (c)q.
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REFERENCES

References: 

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