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On a variety of commutative multiplicatively idempotent semirings

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Abstract

We prove that the variety \({{\mathscr {V}}}\) of commutative multiplicatively idempotent semirings satisfying \(x+y+xyz\approx x+y\) is generated by a single three-element semiring. Moreover, we describe a normal form system for terms in \({{\mathscr {V}}}\) and we show that the word problem in \({{\mathscr {V}}}\) is solvable. Although \({{\mathscr {V}}}\) is locally finite, it is residually big.

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References

  1. Bergman, C.: Universal Algebra: Fundamentals and Selected Topics. Taylor & Francis, Boca Raton (2012)

    MATH  Google Scholar 

  2. Chajda, I., Länger, H.: Subdirectly irreducible commutative multiplicatively idempotent semirings. Algebra Univ. (to appear)

  3. Chajda, I., Länger, H., Švrček, F.: Multiplicatively idempotent semirings. Math. Bohem. 140, 35–42 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Golan, J.S.: Semirings and Their Applications. Kluwer, Dordrecht (1999)

    Book  MATH  Google Scholar 

  5. Guzmán, F.: The variety of Boolean semirings. J. Pure Appl. Algebra 78, 253–270 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kuich, W., Salomaa, A.: Semirings, Automata, Languages. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  7. Pastijn, F., Zhao, X.Z.: Varieties of idempotent semirings with commutative multiplication (unpublished)

  8. Vechtomov, E.M., Petrov, A.A.: Multiplicatively idempotent semirings. J. Math. Sci. 206, 634–653 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Support of the research by the Austrian Science Fund (FWF), Project I 1923-N25, and the Czech Science Foundation (GAČR), Project 15-34697L, as well as by AKTION Austria - Czech Republic, Project 75p11, is gratefully acknowledged. We would like to thank the anonymous referees for their helpful comments and suggestions for the improvement of this paper.

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Correspondence to Helmut Länger.

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Communicated by Mikhail Volkov.

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Chajda, I., Länger, H. On a variety of commutative multiplicatively idempotent semirings. Semigroup Forum 94, 610–617 (2017). https://doi.org/10.1007/s00233-016-9786-9

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  • DOI: https://doi.org/10.1007/s00233-016-9786-9

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