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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regular identities in lattices

Author: R. Padmanabhan
Journal: Trans. Amer. Math. Soc. 158 (1971), 179-188
MSC: Primary 06.30
MathSciNet review: 0281661
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Abstract: An algebraic system $ \mathfrak{A} = \langle A; + , \circ \rangle $ is called a quasilattice if the two binary operations + and $ ^\circ$ are semilattice operations such that the natural partial order relation determined by + enjoys the substitution property with respect to $ ^\circ$ and vice versa. An identity ``$ f = g$'' in an algebra is called regular if the set of variables occurring in the polynomial $ f$ is the same as that in $ g$. It is called $ n$-ary if the number of variables involved in it is at the most $ n$. In this paper we show that the class of all quasilattices is definable by means of ternary regular lattice identities and that these identities span the set of all regular lattice identities and that the arity of these defining equations is the best possible. From these results it is deduced that the class of all quasilattices is the smallest equational class containing both the class of all lattices and the class of all semilattices in the lattice of all equational classes of algebras of type $ \langle 2,2\rangle $ and that the lattice of all equational classes of quasilattices is distributive.

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  • [1] Garrett Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, revised edition, American Mathematical Society, New York, N. Y., 1948. MR 0029876
  • [2] G. Grätzer, Universal algebra, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1968.
  • [3] Bjarni Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110–121 (1968). MR 0237402
  • [4] R. Padmanabhan, Studies in the axiomatics of groups and lattices, Doctoral Dissertation, Madurai University, 1967.
  • [5] -, Link-laws in lattice theory, Madurai Univ. J. 1 (1970).
  • [6] Jerzy Płonka, On distributive quasi-lattices, Fund. Math. 60 (1967), 191–200. MR 0216990
  • [7] J. Płonka, On a method of construction of abstract algebras, Fund. Math. 61 (1967), 183–189. MR 0225701
  • [8] J. Płonka, Some remarks on sums of direct systems of algebras, Fund. Math. 62 (1968), 301–308. MR 0233752

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Keywords: Bi-semilattices, quasilattices, Birkhoff systems, regular identities, compatibility property, partition function, equational class, types of algebras, strictly consistent, base
Article copyright: © Copyright 1971 American Mathematical Society