THE LATTICE OF INTERPRETABILITY TYPES OF CANTOR VARIETIES

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Abstract

For integers 1 ≤ m < n, a Cantor variety with m basic n-ary operations ωi and n basic m-ary operations λk is a variety of algebras defined by identities λk(ω1(JOURNAL/allo/04.02/00133313-200443040-00004/OV0335/v/2017-10-07T112350Z/r/image-png),…, ωm(JOURNAL/allo/04.02/00133313-200443040-00004/OV0335/v/2017-10-07T112350Z/r/image-png)) = xk and ωi(λ1(JOURNAL/allo/04.02/00133313-200443040-00004/OV0451/v/2017-10-07T112350Z/r/image-png),…, λn(JOURNAL/allo/04.02/00133313-200443040-00004/OV0451/v/2017-10-07T112350Z/r/image-png)) = yi, where JOURNAL/allo/04.02/00133313-200443040-00004/OV0335/v/2017-10-07T112350Z/r/image-png = (x1,…, xn) and JOURNAL/allo/04.02/00133313-200443040-00004/OV0451/v/2017-10-07T112350Z/r/image-png = (y1,…, ym). We prove that interpretability types of Cantor varieties form a distributive lattice, 𝒞, which is dual to the direct product 𝒵1 × 𝒵2 of a lattice, 𝒵1, of positive integers respecting the natural linear ordering and a lattice, 𝒵2, of positive integers with divisibility. The lattice 𝒞 is an upper subsemilattice of the lattice ℒint of all interpretability types of varieties of algebras.

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