Timeline for When are the congruence lattices nicer?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 29 at 14:55 | history | edited | Jukka Kohonen |
tag fix (order lattices)
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| Aug 26, 2014 at 21:53 | vote | accept | Noah Schweber | ||
| Aug 25, 2014 at 4:19 | answer | added | The Masked Avenger | timeline score: 3 | |
| Aug 25, 2014 at 1:31 | comment | added | Noah Schweber | Okay, now I follow (I didn't know that those were the only two varieties of distributive lattices). If you post this as an answer, I'll accept it. Thanks! | |
| Aug 25, 2014 at 1:02 | comment | added | The Masked Avenger | Actually, I should know it: a congruence lattice is an algebraic lattice, and the class of algebraic lattices is not a lattice variety. So removing the varietal component of ' and studying its iterates reduces to seeing how ' acts on algebraic lattices. | |
| Aug 25, 2014 at 0:52 | comment | added | The Masked Avenger | For question 1, the answer for strict containment is that L should contain a lattice that is not distributive. Again, one thing I don't know: if C is the class of congruence lattices of a variety of algebras, is C also a lattice variety? There are some forbidden lattice results, but I suspect they are restricted to finite algebras avoiding having certain nondistributive lattices as congruence lattices. | |
| Aug 25, 2014 at 0:45 | comment | added | The Masked Avenger | There are only two varieties of distributive lattices: the variety generated by the two element lattice, and the trivial variety generated by the one element lattice. All other lattice varieties contain the variety of distributive lattices. See e.g. Algebras, Lattices, Varieties Vol. 1 by Taylor, McNulty and McKenzie for proofs. | |
| Aug 24, 2014 at 23:57 | comment | added | Noah Schweber | I'm sorry, I'm still not following; I think I'm just being slow: given that $V'$ is a lattice of distributive varieties, why can we conclude that $V''$ is either trivial or generated by 2? (Or is it known that these are the only two varieties of distributive lattices? That seems odd.) Also, regardless of this, question 1 still seems potentially interesting. | |
| Aug 24, 2014 at 23:39 | comment | added | The Masked Avenger | Because any nontrivial lattice will have its congruence lattice be nontrivial. You will have V' be some lattice variety, then V'' will either be trivial or generated by the two element lattice. More ''s does not change that, if ' means variety generated by congruence lattices. | |
| Aug 24, 2014 at 23:35 | comment | added | The Masked Avenger | If you change ' so that it is the class of congruence lattices and not the variety generated by congruence lattices, you may bump into an open problem. It may be known whether any distr. lattice is the congruence lattice of a distributive lattice but I don't know that. | |
| Aug 24, 2014 at 23:31 | comment | added | Noah Schweber | I didn't know about that result, that's nice! But I don't see why that implies that ' stabilizes. | |
| Aug 24, 2014 at 23:29 | comment | added | The Masked Avenger | 1942 result: lattices are a congruence distributive variety; thus V'' stabilizes under '. | |
| Aug 24, 2014 at 22:50 | history | asked | Noah Schweber | CC BY-SA 3.0 |