半格的可置换同余格的同时表示
Simultaneous representations of semilattices by lattices with permutable congruences
摘要：
同余格问题（CLP）是R. P. Dilworth在40年代提出了一个问题：是否每个分配{，0}-半格S同构于格L的紧同余的半格Conc L。虽然这个问题仍然是开放的，许多部分的解决方案已经获得，积极的和消极的。已知CLP的解对于所有S都是正的，使得$|S|\leq\aleph\_1$。此外，可以用置换同余取L。这与$|S| \geq\aleph\_2$的情况形成对比，其中存在反例S，对于反例S，例如，L不能被分段补。本文证明了这些反例的格也不可能有置换同余。我们还隔离尼特，这些结果的组合类似物。我们得到的所有“nite”语句都是Conc函子的合并性质。最强的已知正结果，起源于第一作者的早期工作，意味着许多由平方2^2索引的半格的图可以关于Conc函子提升。我们证明了后面的结果不能推广到立方体2^3。特别是，我们给出了一个例子的立方体图的nite布尔半格和半格嵌入，不能解除，相对于Conc函子，格置换同余。我们还将我们的许多结果推广到几乎可置换同余格，即$\ga\jj\gb=\ga\gb\uu\gb\ga$，对所有的同余a和b。最后给出了一个简短的证明，证明了从nite布尔半格到格的函子都不能提升nite布尔半格上的Conc函子。
英文摘要：
The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilatticeS is isomorphic to the semilattice Conc L of compact congruences of a lattice L. While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that $|S|\leq\aleph\_1$. Furthermore, one can then take L with permutable congruences. This contrasts with the case where $|S| \geq\aleph\_2$, where there are counterexamples S for which L cannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well. We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Conc functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square 2^2 can be lifted with respect to the Conc functor. We prove that the latter results cannot be extended to the cube, 2^3. In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Conc functor, by lattices with permutable congruences. We also extend many of our results to lattices with almost permutable congruences, that is, $\ga\jj\gb=\ga\gb\uu\gb\ga$, for all congruences a and b. We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Conc functor on finite Boolean semilattices.
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