Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Compact Representation for Modular Semilattices and its Applications   A modular semilattice is a semilattice generalization of a modular lattice.
We establish a Birkhoff-type representation theorem for modular semilattices,
which says that every modular semilattice is isomorphic to the family of ideals
in a certain poset with additional relations.This new poset structure, which we
axiomatize in this paper, is called a PPIP (projective poset with inconsistent
pairs). A PPIP is a common generalization of a PIP (poset with inconsistent
pairs) and a projective ordered space. The former was introduced by
Barth\'elemy and Constantin for establishing Birkhoff-type theorem for median
semilattices, and the latter by Herrmann, Pickering, and Roddy for modular
lattices. We show the $\Theta (n)$ representation complexityand a construction
algorithm for PPIP-representations of $(\wedge, \vee)$-closed sets in the
product $L^n$ of modular semilattice $L$. This generalizes the results of Hirai
and Oki for a special median semilattice $S_k$. We also investigate
implicational bases for modular semilattices. Extending earlier results of Wild
and Herrmann for modular lattices, we determine optimal implicational bases and
develop a polynomial time recognition algorithm for modular semilattices. These
results can be applied to retain the minimizer set of a submodular function on
a modular semilattice.

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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: cs