Abstract
An implication semigroup is an algebra of type (2, 0) with a binary operation → and a 0-ary operation 0 satisfying the identities \((x\rightarrow y)\rightarrow z\approx x\rightarrow (y\rightarrow z)\), \((x\rightarrow y)\rightarrow z\approx \left [(z^{\prime }\rightarrow x)\rightarrow (y\rightarrow z)'\right ]'\) and \(0^{\prime \prime }\approx 0\) where \(\mathbf {u}^{\prime }\) means \(\mathbf u\rightarrow 0\) for any term u. We completely describe the lattice of varieties of implication semigroups. It turns out that this lattice is non-modular and consists of 16 elements.
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Acknowledgments
This work was started when the second and the third authors took part in the Emil Artin International Conference held in Yerevan in May–June of 2018. The authors are deeply grateful to Professor Yuri Movsisyan and his colleagues for the excellent organization of the conference and the creation of the favorable atmosphere that contributed to the appearance of this article. The authors would like to express also their gratitude to the anonymous referee for his/her valuable remarks and suggestions that contributed to a significant improvement of the original version of the manuscript, in general and to the current proof of Theorem 1.1, in particular.
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The first and the third authors were partially supported by the Ministry of Education and Science of the Russian Federation (project 1.6018.2017/8.9) and by the Russian Foundation for Basic Research (grant No. 17-01-00551).
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Gusev, S.V., Sankappanavar, H.P. & Vernikov, B.M. The Lattice of Varieties of Implication Semigroups. Order 37, 271–277 (2020). https://doi.org/10.1007/s11083-019-09503-5
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DOI: https://doi.org/10.1007/s11083-019-09503-5