Abstract
We show that an idempotent variety has a d-dimensional cube term if and only if its free algebra on two generators has no d-ary compatible cross. We employ Hall’s Marriage Theorem to show that an idempotent variety \({\mathcal{V}}\) of finite signature whose fundamental operations have arities n 1, . . . , n k, has a d-dimensional cube term for some d if and only if it has one of dimension \({1 + \sum_{i=1}^{k} (n_{i} - 1)}\). This upper bound on the dimension of a minimal-dimension cube term for \({\mathcal{V}}\) is shown to be sharp. We show that a pure cyclic term variety has a cube term if and only if it contains no 2- element semilattice. We prove that the Maltsev condition “existence of a cube term” is join prime in the lattice of idempotent Maltsev conditions.
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Presented by J. Berman.
This material is based upon work supported by the National Science Foundation grant no. DMS 1500254 and the Hungarian National Foundation for Scientific Research (OTKA) grant no. K104251 and K115518.
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Kearnes, K.A., Szendrei, Á. Cube term blockers without finiteness. Algebra Univers. 78, 437–459 (2017). https://doi.org/10.1007/s00012-017-0476-6
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DOI: https://doi.org/10.1007/s00012-017-0476-6