Abstract
We define an identity ε to be hypersatisfied by a variety V if, whenever the operation symbols of V, are replaced by arbitrary terms (of appropriate arity) in the operations of V, the resulting identity is satisfied by V in the usual sense. Whenever the identity ε is hypersatisfied by a variety V, we shall say that ε is a V hyperidentity. For example, the identity x + x ⋅ y = x ⋅(x + y) is hypersatisfied by the variety L of all lattices. A proof of this consists of a case-by-case examination of { + , ⋅} {x, y, x ∨ y, x ∧ y}, the set of all binary lattice terms. In an earlier work, we exhibited a hyperbase ΣL for the set of all binary lattice (or, equivalently, quasilattice) hyperidentities of type 2, 2. In this paper we provide a greatly refined hyperbase Σ L . The proof that Σ L is a hyperbase was obtained by using the automated reasoning program Otter 3.0.4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczél, J.: Proof of a theorem on distributive hyperidentities, Algebra Universalis 1 (1971), 1-6.
Belousov, D.: Systems of quasigroups with generalized identities, Uspekhi Mat. Nauk 20 (1965).
Denecke, K. and Reichel, M.: Monoids of hypersubstitutions and M-solid varieties, in: Contributions General Algebra 9, Hölder-Pichler-Tempsky, Vienna, Teubner, Stuttgart, 1995, pp. 117-126.
Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices, W. H. Freeman & Co, San Francisco, 1971.
Grätzer, G.: Universal Algebra, 2nd edn, Springer-Verlag, New York, 1979.
Graczynska, E. and Schweigert, D.: Hyperidentities of a given type, Algebra Universalis 27 (1990), 305-318.
McCune, W.: OTTER 3.0 reference manual and guide, Tech. Report ANL-94/6, Argonne National Laboratory, Argonne, IL, 1994.
Padmanabhan, R.: Regular identities in lattices, Trans. Amer. Math. Soc. 158(1) (1971), 179-188.
Padmanabhan, R. and Penner, P.: Binary hyperidentities of lattices, Aequationes Math. 44 (1992), 154-167.
Padmanabhan, R. and Penner, P.: Subdirectly irreducible n-quasilattices, Algebra Universalis 31 (1994), 124-134.
Penner, P.: Hyperidentities of Lattices, Semilattices and Diagonal Semigroups, Ph.D. thesis, University of Manitoba.
Taylor, W.: Hyperidentities and hypervarieties, Aequationes Math. 23 (1981), 30-49.
Rights and permissions
About this article
Cite this article
Padmanabhan, R., Penner, P. A Hyperbase for Binary Lattice Hyperidentities. Journal of Automated Reasoning 24, 365–370 (2000). https://doi.org/10.1023/A:1006279127474
Issue Date:
DOI: https://doi.org/10.1023/A:1006279127474