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The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\)

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Abstract

The aim of this paper is to study the varieties of semilattice-ordered Burnside semigroups satisfying \(x^3\approx x\) and \(xy\approx yx.\) It is shown that the collection of all such varieties forms a distributive lattice of order 9. Also, all of them are finitely based and finitely generated. This gives a generalization and expansion of the results obtained by McKenzie and Romanowska (Contrib Gen Algebra Proc Klagenf Conf 1978 1:213–218, 1979).

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Acknowledgments

The authors thank the anonymous referee for their valuable comments and suggestions which lead to the final version of this paper. The authors also thank Dr. Yong Shao for discussions contributed to this paper. This work is supported by National Natural Science Foundation of China (11571278), Miaomiao Ren is supported by Natural Science Foundation of Shannxi Province (2015JQ1210), Xianzhong Zhao is supported by National Natural Science Foundation of China (11261021) and Grant of Natural Science Foundation of Jiangxi Province (20142BAB201002).

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Correspondence to Xianzhong Zhao.

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In loving memory of Miaomiao’s grandmother Xiulan Li.

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Ren, M., Zhao, X. The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\) . Period Math Hung 72, 158–170 (2016). https://doi.org/10.1007/s10998-016-0116-5

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