Abstract
A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclonelin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclonelin(τ, τ′).
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References
Malcev A. I., Algebraic Systems, Akademie-Verlag, Berlin (1973).
Denecke K. and Phusanga D., “Hyperformulas and solid algebraic systems,” Stud. Log., vol. 90, 263–268 (2000).
Lawvere F. W., Functorial Semantics of Algebraic Theories: Diss., Columbia Univ., New York (1963).
Denecke K. and Phusanga D., “Hypersatisfaction of formulas in algebraic systems,” Discuss. Math. Gen. Algebra Appl., vol. 29, no. 2, 123–152 (2009).
Denecke K., “The partial clone of linear terms,” Sib. Math. J., vol. 57, no. 4, 589–598 (2016).
Pilitowska A., “Linear identities in graph algebras,” Comment. Math. Univ. Carolin., vol. 50, 11–24 (2009).
Grätzer G. and Lakser H., “Identities for globals (complex algebras) of algebras,” Colloq. Math., vol. 56, no. 1, 19–29 (1988).
Denecke K. and Wismath S. L., Hyperidentities and Clones, Gordon and Breach, New York (2000).
Changphas Th., Denecke K., and Pibaljommee B., Linear Terms and Linear Hypersubstituions [Preprint], KhonKaen (2014).
Koppitz J. and Denecke K., M-Solid Varieties of Algebras, Springer-Verlag, New York (2006).
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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 734–750.
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Denecke, K. The Partial Clone of Linear Formulas. Sib Math J 60, 572–584 (2019). https://doi.org/10.1134/S0037446619040037
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DOI: https://doi.org/10.1134/S0037446619040037