Content deleted Content added
DataTrekker (talk | contribs) Link suggestions feature: 3 links added. Tags: Visual edit Mobile edit Mobile web edit Newcomer task Suggested: add links |
Link suggestions feature: 3 links added. Tags: Visual edit Mobile edit Mobile web edit Newcomer task Suggested: add links |
||
| (One intermediate revision by one other user not shown) | |||
Line 27:
Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice.
It is important to observe that the solution lattice found in Grätzer and Schmidt's proof is ''sectionally complemented'', that is, it has a [[Greatest element and least element|least element]] (true for any finite lattice) and for all elements ''a'' ≤ ''b'' there exists an element ''x'' with ''a'' ∨ ''x'' = ''b'' and ''a'' ∧ ''x'' = ''0''. It is also in that paper that CLP is first stated in published form, although it seems that the earliest attempts at CLP were made by Dilworth himself. Congruence lattices of finite lattices have been given an enormous amount of attention, for which a reference is Grätzer's 2005 monograph.{{sfn|Grätzer|2005}}
----
Line 66:
'''Theorem.'''
The [[functor]] Con<sub>c</sub>, defined on all algebras of a given [[signature (logic)|signature]], to all (∨,0)-semilattices, [[Limit (category theory)|preserves direct limits]].
==Schmidt's approach via distributive join-homomorphisms==
Line 229:
</math>
(Observe the faint formal similarity with [[Resolution (logic)|first-order resolution]] in [[mathematical logic]]. Could this analogy be pushed further?)
The proof of the theorem above runs by setting a ''structure'' theorem for congruence lattices of semilattices—namely, the Erosion Lemma, against ''non-structure'' theorems for free distributive extensions ''G(Ω)'', the main one being called the ''Evaporation Lemma''. While the latter are technically difficult, they are, in some sense, predictable. Quite to the contrary, the proof of the Erosion Lemma is elementary and easy, so it is probably the strangeness of its statement that explains that it has been hidden for so long.
Line 741:
| pages=321–356
| doi=10.1142/S0218196708004469| s2cid=7674741
| arxiv=math/0601058
}}
* {{cite journal
Line 750 ⟶ 751:
| date=2007
| pages=610–625
| doi=10.1016/j.aim.2007.05.016 | doi-access=free
}}
[[Category:Lattice theory]]
| |||