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• W3085082021 abstract "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 Barthelemy and Constantin for establishing Birkhoff-type theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the Θ(n) representation complexity and a construction algorithm for PPIP-representations of (∧,∨)-closed sets in the product Ln of modular semilattice L. This generalizes the results of Hirai and Oki for a special median semilattice Sk. 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." @default.
• W3085082021 created "2020-09-21" @default.
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• W3085082021 date "2020-09-10" @default.
• W3085082021 modified "2023-10-03" @default.
• W3085082021 title "A Compact Representation for Modular Semilattices and Its Applications" @default.
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• W3085082021 doi "https://doi.org/10.1007/s11083-019-09516-0" @default.
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