Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of algebraic distributive lattices with $\aleph\_1$
  compact elements as ideal lattices of regular rings   We prove the following result: Theorem. Every algebraic distributive lattice
D with at most $\aleph\_1$ compact elements is isomorphic to the ideal lattice
of a von Neumann regular ring R. (By earlier results of the author, the
$\aleph\_1$ bound is optimal.) Therefore, D is also isomorphic to the
congruence lattice of a sectionally complemented modular lattice L, namely, the
principal right ideal lattice of R. Furthermore, if the largest element of D is
compact, then one can assume that R is unital, respectively, that L has a
largest element. This extends several known results of G.M. Bergman, A.P. Huhn,
J. T&#730;uma, and of a joint work of G. Gr\"{a}tzer, H. Lakser, and the
author, and it solves Problem 2 of the survey paper [10]. The main tool used in
the proof of our result is an amalgamation theorem for semilattices and
algebras (over a given division ring), a variant of previously known
amalgamation theorems for semilattices and lattices, due to J. Tuma, and G.
Gr\"{a}tzer, H. Lakser, and the author.

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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg