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Varieties of Birkhoff Systems Part I

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Abstract

A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bottom part of the lattice of subvarieties of Birkhoff systems, in particular the role of meet and join distributive Birkhoff systems. Our purpose in this note is to further explore the lattice of subvarieties of Birkhoff systems. A primary tool is consideration of splittings and finite bichains, Birkhoff systems whose join and meet reducts are both chains. We produce an infinite family of subvarieties of Birkhoff systems generated by finite splitting bichains, and describe the poset of these subvarieties. Consideration of these splitting varieties also allows us to considerably extend knowledge of the lower part of the lattice of subvarieties of Birkhoff systems

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

    John Harding

  2. Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662, Warsaw, Poland

    Anna B. Romanowska

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  1. John Harding
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  2. Anna B. Romanowska
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Correspondence to John Harding.

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Harding, J., Romanowska, A.B. Varieties of Birkhoff Systems Part I. Order 34, 45–68 (2017). https://doi.org/10.1007/s11083-016-9388-x

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