Abstract
The concept of implicit operation on pseudovarieties of semigroups goes back to Eilenberg and Schutzenberger [1]. The author in [2–5] generalized this concept to other classes of algebras and established a connection between these operations and positively conditional termal functions in the case of uniform local finiteness of the algebras of the class in question. In this article we put forth the concept of an implicit operation for an arbitrary universal algebra, not necessarily locally finite, and establish a connection between these operations and infinite analogs of positively conditional terms, as well as ∞-quasi-identities arising in the algebraic geometry of universal algebras. We also consider conditions for implicit equivalence of algebras to lattices, semilattices, and Boolean algebras.
Similar content being viewed by others
References
Eilenberg S. and Schutzenberger M. P., “On pseudovarieties,” Adv. Math., 19, No. 3, 413–418 (1976).
Pinus A. G., “Implicit operations on the categories of universal algebras,” Siberian Math. J., 50, No. 1, 117–122 (2009).
Pinus A. G., “On the implicit conditional operations defined on pseudouniversal classes,” J. Math. Sci. (New York), 140, No. 2, 295–302 (2007).
Pinus A. G., “Positively conditional pseudovarieties and implicit operations on them,” Siberian Math. J., 47, No. 2, 307–314 (2006).
Pinus A. G., “∃+-conditional varieties, ∃+-conditional pseudovarieties, and implicit operations on them,” in: Algebra and Model Theory. 5 [in Russian], Novosibirsk State Technical Univ., Novosibirsk, 2005, pp. 138–161.
Pinus A. G., “Geometric scales for varieties of algebras and quasi-identities,” Siberian Adv. in Math., 20, No. 3, 217–222(2010).
Pinus A. G., “On ∞-quasivarieties,” Russian Math. (Izv. VUZ. Mat.), 55, No. 8, 33–37 (2011).
Pinus A. G., “Conditional identity calculus and the conditioned rational equivalence,” Algebra and Logic, 37, No. 4, 245–259 (1998).
Plotkin B., “Varieties of algebras and algebraic varieties,” Israel Math. J., 96, No. 2, 511–522 (1996).
Plotkin B. I., “Some concepts of algebraic geometry in universal algebra,” St. Petersburg Math. J., 9, No. 4, 859–879(1998).
Plotkin B. I., “Algebras with the same (algebraic) geometry,” Proc. Steklov Inst. Math., 242, 165–196 (2003); in: Mathematical Logic and Algebra: Proceedings Dedicated to P. S. Novikov on the Occasion of His 100 Birthday, Nauka, Moscow, 2003, Vol. 242, pp. 176–207.
Malcev A. I., “Structure characterization of some classes of algebras,” Dokl. Akad. Nauk SSSR, 120,No. 1, 29–32 (1958).
Pinus A. G., Conditional Terms and Their Applications to Algebra and the Theory of Computations [in Russian], Izdat. NGTU, Novosibirsk (2002).
Pinus A. G., “On functions commuting with semigroups of transformations of algebras,” Siberian Math. J., 41, No. 6, 1166–1173 (2000).
Pinus A. G., Derived Structures of Universal Algebras [in Russian], Izdat. NGTU, Novosibirsk (2007).
Pinus A. G., “Galois-correspondences between implicit operations and categories of universal algebras,” Vestnik NGU, 11, No. 3, 148–154 (2011).
Pinus A. G., “On algebras conditionally rationally equivalent to semilattices and Boolean algebras,” Siberian Math. J., 39, No. 1, 106–112 (1998).
Pinus A. G., “New algebraic invariants for definable subsets in universal algebra,” Algebra and Logic, 50, No. 2, 146–160 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2012 Pinus A.G.
__________
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 5, pp. 1077–1090, September–October, 2012.
Rights and permissions
About this article
Cite this article
Pinus, A.G. Implicitly equivalent universal algebras. Sib Math J 53, 862–871 (2012). https://doi.org/10.1134/S0037446612050114
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446612050114