Abstract
Skew lattices form a class of non-commutative lattices. Spinks' Theorem [Matthew Spinks, On middle distributivity for skew lattices, Semigroup Forum 61 (2000), 341-345] states that for symmetric skew lattices the two distributive identities \(x\wedge (y\vee z)\wedge x=(x\wedge y\wedge x)\vee (x\wedge z\wedge x)\) and \(x\vee (y\wedge z)\vee x=(x\vee y\vee x)\wedge (x\vee z\vee x)\) are equivalent. Up to now only computer proofs of this theorem have been known. In the present paper the author presents a direct proof of Spinks' Theorem. In addition, a new result is proved showing that the assumption of symmetry can be omitted for cancellative skew lattices.
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Cvetko-Vah, K. A New Proof of Spinks' Theorem. Semigroup Forum 73, 267–272 (2006). https://doi.org/10.1007/s00233-006-0603-8
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DOI: https://doi.org/10.1007/s00233-006-0603-8