Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simultaneous representations of semilattices by lattices with permutable
  congruences   The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the
forties, asks whether every distributive {&#8744;, 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
&#64257;nite, combinatorial analogues of these results. All the "&#64257;nite"
statements that we obtain are amalgamation properties of the Conc functor. The
strongest known positive results, which originate in earlier work by the
&#64257;rst 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 &#64257;nite 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 &#64257;nite Boolean semilattices to lattices can lift the Conc
functor on &#64257;nite Boolean semilattices.

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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg