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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
DOI: https://doi.org/10.1090/S0002-9947-1971-0281661-3
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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DOI: https://doi.org/10.1090/S0002-9947-1971-0281661-3
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

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