Cube term blockers without finiteness |
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Authors: | Keith A Kearnes Ágnes Szendrei |
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Institution: | 1.Department of Mathematics,University of Colorado,Boulder,USA |
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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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