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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero


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CiteScore 2018: 0.47
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Volume 48, Issue 4

Issues

Weakly Idempotent Lattices and Bilattices, Non-Idempotent Plonka Functions

D. S. Davidova / Yu. M. Movsisyan
Published Online: 2015-12-04 | DOI: https://doi.org/10.1515/dema-2015-0037

Abstract

In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear.

Keywords: weakly idempotent semilattice; weakly idempotent lattice; weakly idempotent bilattice; interlaced operation; interlaced weakly idempotent bilattice; hyperidentity; non-idempotent Plonka function; weakly Plonka sum; weakly idempotent quasilattice

References

  • [1] A. D. Anosov, On homomorphisms of many-sorted algebraic systems in connection with cryptographic applications, Discrete Math. Appl. 17(4) (2007), 331-347.CrossrefGoogle Scholar

  • [2] B. H. Arnold, Distributive lattices with a third operation defined, Pacific J. Math. 1 (1951), 33-41.Google Scholar

  • [3] A. Avron, The structure of interlaced bilattices, Math. Structures Comput. Sci. 6 (1996), 287-299.Google Scholar

  • [4] G. M. Bergman, An Invitation on General Algebra and Universal Constructions, Second Edition, Springer, 2015.Google Scholar

  • [5] F. Bou, R. Jansana, U. Rivieccio, Varieties of interlaced bilattices, Algebra Universalis 66 (2011), 115-141.CrossrefWeb of ScienceGoogle Scholar

  • [6] A. Craig, L. M. Cabrer, H. A. Priestley, Beyond FOUR: representations of noninterlaced bilattices using natural duality, Research Workshop on Duality Theory in Algebra, Logic and Computer Science, University of Oxford, 16-16, June 13-14, 2012.Google Scholar

  • [7] B. A. Davey, The product representation theorem for interlaced pre-bilattices: some historical remarks, Algebra Universalis 70 (2013), 403-409.Web of ScienceCrossrefGoogle Scholar

  • [8] K. Denecke, J. Koppitz, M-Solid Varieties of Algebras. Advances in Mathematic, 10, Spriger-Science + Business Media, New York, 2006.Google Scholar

  • [9] K. Denecke, S. L. Wismath, Hyperidentities and Clones, Gordon and Breach Science Publishers, 2000.Google Scholar

  • [10] M. Fitting, Bilattices and the theory of truth, J. Philos. Logic 18 (1989), 225-256.Google Scholar

  • [11] M. Fitting, Bilattices in logic programming, in.: G. Epstein (ED.), 20th International Symposium on Multiple-Valued Logic, 238-246, IEEE Press, 1990.Google Scholar

  • [12] M. Fitting, Bilattices and the semantics of logic programming, J. Logic Programming 11 (1991), 91-116.Google Scholar

  • [13] E. Fried, G. Gratzer, A nonassocaative extension of the class of distributive lattices, Pacific J. Math. 49(1) (1973), 59-78.Google Scholar

  • [14] E. Fried, Weakly associative lattices with congruence extension property, Algebra Universalis 4 (1974), 151-162.CrossrefGoogle Scholar

  • [15] G. Gargov, Knowledge, uncertainty and ignorence in logic: bilattices and beyound, J.Appl. Non-Classical Logics 9 (1999), 195-283.Google Scholar

  • [16] G. Gratzer, Universal Algebra, Springer-Verlag, 2008.Google Scholar

  • [17] M. L. Ginsberg, Multi-valued logics: a uniform approach to reasoning in artificial intelligence, Computational Intelligence 4 (1988), 265-316.CrossrefGoogle Scholar

  • [18] E. Graczynska, On normal and regular identities, Algebra Universalis 27 (1990), 387-397.CrossrefGoogle Scholar

  • [19] E. Graczynska, Algebra of M-Solid Quasilattices, Siatras International Bookshop, Athens, 2014.Google Scholar

  • [20] J. Jakubik, M. Kolibiar, On some properties of a pair of lattices, Czechoslovak Math. J. 4 (1954), 1-27.Google Scholar

  • [21] J. Jakubik, M. Kolibiar, Lattices with a third distributive operation, Math. Slovaca 27 (1977), 287-292.Google Scholar

  • [22] A. Jung, U. Rivieccio, Priestley duality for bilattices, Studia Logica 100 (2012), 223-252.Google Scholar

  • [23] S. A. Kiss, Transformations on Lattices and Structures of Logic, New York, 1947.Google Scholar

  • [24] I. I. Melnik, Nilpotent shift of manifolds, Math. Notes 14 (1973), 387-397.Google Scholar

  • [25] Yu. M. Movsisyan, Bilattices and hyperidentities, Proceedings of the Steclov Institute of Mathematics 274 (2011), 174-192Google Scholar

  • [26] Yu. M. Movsisyan, Interlaced, modular, distributive and Boolean bilattices, Armenian J. Math. 1(3) (2008), 7-13.Google Scholar

  • [27] Yu. M. Movsisyan, Introduction to the Theory of Algebras with Hyperidentities, Yerevan State University Press, Yerevan, 1986. (Russian)Google Scholar

  • [28] Yu. M. Movsisyan, Hyperidentitties in algebras and varieties, Uspekhi Mat. Nauk 53(1) (1998), 61-114, (Russian). English transl. in Russian Math. Surveys 53 (1998), 57-108.Google Scholar

  • [29] Yu. M. Movsisyan, Hyperidentities and hypervarieties, Sci. Math. Jpn. 54(3) (2001), 595-640.Google Scholar

  • [30] Yu. M. Movsisyan, Hyperidentities of Boolean algebras, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 654-672, (Russian). English transl. in Russian Acad. Sci. Izv. Math. 40 (1993), 607-622.Google Scholar

  • [31] Yu. M. Movsisyan, Algebras with hyperidentities of the variety of Boolean algebras, Izv. Russ. Acad. Nauk. Ser. Mat. 60 (1996), 127-168. English transl. in Russian Acad. Sci. Izv. Math. 60 (1996), 1219-1260.Google Scholar

  • [32] Yu. M. Movsisyan, V. A. Aslanyan, Hyperidentities of De Morgan algebras, Log. J. IGPL 20 (2012), 1153-1174.Google Scholar

  • [33] Yu. M. Movsisyan, A. B. Romanowska, J. D. H. Smith, Superproducts, hyperidentities, and algebraic structures of logic programming, J. Combin. Math. Combin. Comput. 58 (2006), 101-111.Google Scholar

  • [34] R. Padamanabhan, P. Penner, A hyperbase for binary lattice hyperidentities, J. Automat. Reason. 24 (2000), 365-370.Google Scholar

  • [35] J. Plonka, On a method of construction of abstract algebras, Fund. Math. 61 (1967), 183-189.Google Scholar

  • [36] J. Plonka, On varieties of algebras defined by identities of some special forms, Houston J. Math. 14 (1988), 253-263Google Scholar

  • [37] J. Plonka, A. Romanowska, Semilattice sums, Universal Algebra and Quasigroup Theory, Helderman Verlag, Berlin, 1992, 123-158.Google Scholar

  • [38] A. P. Pynko, Regular bilattices, J. Appl. Non-Classical Logics 10 (2000), 93-111.Google Scholar

  • [39] A. Romanowska, J. D. H. Smith, Modes, World Scientific, 2002.Google Scholar

  • [40] H. A. Priestley, Distributive bilattices and their cousins: representation via natural dualities, Research Workshop on Duality Theory in Algebra, Logic and Computer Science, 18-18, University of Oxford, June 13-14, 2012.Google Scholar

  • [41] U. Rivieccio, An algebraic study of bilattice-based logics, PhD Dissertation, University of Barcelona, 2010.Google Scholar

  • [42] A. Romanowska, A. Trakul, On the structure of some bilattices, Universal and Applied Algebra, 235-253, Turawa, 1988. Google Scholar

  • [43] J. D. H. Smith, On groups of hypersubstitutions, Algebra Universalis 64 (2010), 39-48. CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2014-10-02

Revised: 2015-04-20

Published Online: 2015-12-04

Published in Print: 2015-12-01


Citation Information: Demonstratio Mathematica, Volume 48, Issue 4, Pages 509–535, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.1515/dema-2015-0037.

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© by D. S. Davidova. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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