Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Ramsey and the ordering property for classes of lattices and
  semilattices   The class of finite distributive lattices, as many other classes of
structures, does not have the Ramsey property. It is quite common, though, that
after expanding the structures with appropriately chosen linear orders the
resulting class has the Ramsey property. So, one might expect that a similar
result holds for the class of all finite distributive lattices. Surprisingly,
Kechris and Soki\'c have proved in 2012 that this is not the case: no expansion
of the class of finite distributive lattices by linear orders satisfies the
Ramsey property.
  In this paper we prove that the variety of distributive lattices is not an
exception, but an instance of a more general phenomenon. We show that for
almost all nontrivial locally finite varieties of lattices no "reasonable"
expansion of the finite members of the variety by linear orders gives rise to a
Ramsey class. The responsibility for this lies not with the lattices as
structures, but with the lack of algebraic morphisms: if we consider lattices
as partially ordered sets (and thus switch from algebraic embeddings to
embeddings of relational structures) we show that every variety of lattices
gives rise to a class of linearly ordered posets having both the Ramsey
property and the ordering property. It now comes as no surprise that the same
is true for varieties of semilattices.

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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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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: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex