Abstract
Generalizing a linear expression over a vector space, we call a term of an arbitrary type τ linear if its every variable occurs only once. Instead of the usual superposition of terms and of the total many-sorted clone of all terms in the case of linear terms, we define the partial many-sorted superposition operation and the partial many-sorted clone that satisfies the superassociative law as weak identity. The extensions of linear hypersubstitutions are weak endomorphisms of this partial clone. For a variety V of one-sorted total algebras of type τ, we define the partial many-sorted linear clone of V as the partial quotient algebra of the partial many-sorted clone of all linear terms by the set of all linear identities of V. We prove then that weak identities of this clone correspond to linear hyperidentities of V.
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Dedicated to the memory of H.-J. Hoehnke on occasion of his 90th birthday in October 2015.
Original Russian Text Copyright © 2016 Denecke K.
Potsdam; KhonKaen. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 4, pp. 755–767, July–August, 2016; DOI: 10.17377/smzh.2016.57.403. Original article submitted March 13, 2015.
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Denecke, K. The partial clone of linear terms. Sib Math J 57, 589–598 (2016). https://doi.org/10.1134/S0037446616040030
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DOI: https://doi.org/10.1134/S0037446616040030