54 research outputs found
Shard polytopes
For any lattice congruence of the weak order on permutations, N. Reading
proved that gluing together the cones of the braid fan that belong to the same
congruence class defines a complete fan, called a quotient fan, and V. Pilaud
and F. Santos showed that it is the normal fan of a polytope, called a
quotientope. In this paper, we provide a simpler approach to realize quotient
fans based on Minkowski sums of elementary polytopes, called shard polytopes,
which have remarkable combinatorial and geometric properties. In contrast to
the original construction of quotientopes, this Minkowski sum approach extends
to type .Comment: 73 pages, 35 figures; Version 2: minor corrections for final versio
Distributive Lattices have the Intersection Property
Distributive lattices form an important, well-behaved class of lattices. They
are instances of two larger classes of lattices: congruence-uniform and
semidistributive lattices. Congruence-uniform lattices allow for a remarkable
second order of their elements: the core label order; semidistributive lattices
naturally possess an associated flag simplicial complex: the canonical join
complex. In this article we present a characterization of finite distributive
lattices in terms of the core label order and the canonical join complex, and
we show that the core label order of a finite distributive lattice is always a
meet-semilattice.Comment: 9 pages, 3 figures. Final version. Comments are very welcom
Meet-distributive lattices have the intersection property
summary:This paper is an erratum of H. Mühle: Distributive lattices have the intersection property, Math. Bohem. (2021). Meet-distributive lattices form an intriguing class of lattices, because they are precisely the lattices obtainable from a closure operator with the so-called anti-exchange property. Moreover, meet-distributive lattices are join semidistributive. Therefore, they admit two natural secondary structures: the core label order is an alternative order on the lattice elements and the canonical join complex is the flag simplicial complex on the canonical join representations. In this article we present a characterization of finite meet-distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite meet-distributive lattice is always a meet-semilattice
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