The M3[D] construction and n-modularity |
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Authors: | G Grätzer F Wehrung |
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Institution: | (1) Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Manitoba, Canada, e-mail: gratzer@cc.umanitoba.ca, URL http//www.maths.umanitoba.ca/homepages/gratzer.html/ , CA;(2) C.N.R.S., Département de Mathématiques, Université de Caen, Campus II B.P. 5186, F-14032 Caen CEDEX, France, e-mail: wehrung@math.unicaen.fr, URL http//www.math.unicaen.fr/˜wehrung, FR |
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Abstract: | In 1968, Schmidt introduced the M
3D] construction, an extension of the five-element modular nondistributive lattice M
3 by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M
3D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M
n
denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M
3L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M
3L]
M
3Id L]. ? We provide an example of a lattice L such that M
3L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M
3L] construction to an M
4L] construction. This yields, in particular, a bounded modular lattice L such that M
4
L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the
congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice.
Received May 26, 1998; accepted in final form October 7, 1998. |
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Keywords: | and phrases: Lattice modular congruence-preserving extension |
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