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Semiring varieties related to multiplicative Green’s relations on a semiring

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Abstract

In this paper, by means of congruence openings of multiplicative Green’s relations on a semiring we define and study several varieties of semirings, obtain the relationship between these varieties and give Mal’cev product decompositions of some varieties of idempotent semirings. In particular, we establish order embeddings of the lattice of all subvarieties of the variety of multiplicatively idempotent semirings (resp. idempotent semirings) into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice what we introduced.

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Acknowledgements

The authors are particularly grateful to Professors Mikhail V. Volkov and Edmond W. H. Lee for their comments and suggestions contributed to this paper. Yanliang Cheng is supported by National Natural Science Foundation of China (11971383). Yong Shao is supported by National Natural Science Foundation of China (11801239) and Natural Science Foundation of Shaanxi Province (2020JM-425).

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  1. School of Mathematics, Northwest University, Xi’an, 710127, Shaanxi, People’s Republic of China

    Yanliang Cheng & Yong Shao

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  1. Yanliang Cheng
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  2. Yong Shao
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Correspondence to Yong Shao.

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Communicated by Edmond W. H. Lee.

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Cheng, Y., Shao, Y. Semiring varieties related to multiplicative Green’s relations on a semiring. Semigroup Forum 101, 571–584 (2020). https://doi.org/10.1007/s00233-020-10108-3

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