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- W2036341540 abstract "In 1968, Schmidt introduced the M 3[D] 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 $langle x,y,z rangle in D^3$ satisfying $ xwedge y=xwedge z=ywedge z$ . The lattice M 3[D] 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 $ {bfmu}_n$ such that $ {bfmu}_1$ is modularity and ${bfmu}_{n+1}$ is properly weaker than ${bfmu}_n$ . Let M n denote the variety defined by ${bfmu}_n$ , the variety of n-modular lattices. If L is n-modular, then M 3[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M 3[L] $cong $ M 3[Id L]. ¶ We provide an example of a lattice L such that M 3[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product $ Aotimes B$ of two lattices A and B with zero always a lattice. We complement this result by generalizing the M 3[L] construction to an M 4[L] construction. This yields, in particular, a bounded modular lattice L such that M 4 $ otimes $ 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 Grätzer, Lakser, and Schmidt yields a 3-modular lattice." @default.
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- W2036341540 date "1999-05-01" @default.
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- W2036341540 title "The M 3 [ D ] construction and n -modularity" @default.
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- W2036341540 doi "https://doi.org/10.1007/s000120050102" @default.
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