JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (6): 15-22.doi: 10.6040/j.issn.1671-9352.0.2021.135

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Greens relations on a class of semiring which multiplicative reduct is an idempotent semigroup

WANG Jun-ling, SHAO Yong*   

  1. School of Mathematics, Northwest University, Xian 710127, Shaanxi, China
  • Published:2022-06-10

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Abstract: The multiplicatively idempotent semirings satisfying the identities x+x+x=x, 2x+2y=2(x+y) are studied. The characterizations of the binary relations(·overL)∧(+overD),(·overL)∧(+overL),(·overL)∧(+overR),(+overL)∧(·overD)related to the Greens relation of the multiplicative semigroups(additive semigroups)of the semirings are given, and the sufficient and necessary conditions which make these binary relations be congruences are obtained. Moreover, the classes of semirings which are determined by these congruences are proved to be semiring varieties.

Key words: semiring, variety, congruences, Greens relations

CLC Number: 

  • O151.21
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