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On a variety of commutative multiplicatively idempotent semirings

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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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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).

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