14,742 research outputs found
An analogue of distributivity for ungraded lattices
In this paper, we define a property, trimness, for lattices. Trimness is a
not-necessarily-graded generalization of distributivity; in particular, if a
lattice is trim and graded, it is distributive. Trimness is preserved under
taking intervals and suitable sublattices. Trim lattices satisfy a weakened
form of modularity. The order complex of a trim lattice is contractible or
homotopic to a sphere; the latter holds exactly if the maximum element of the
lattice is a join of atoms.
Other than distributive lattices, the main examples of trim lattices are the
Tamari lattices and various generalizations of them. We show that the Cambrian
lattices in types A and B defined by Reading are trim, and we conjecture that
all Cambrian lattices are trim.Comment: 19 pages, 4 figures. Version 2 includes small improvements to
exposition, corrections of typos, and a new section showing that if a group G
acts on a trim lattice by lattice automorphisms, then the sublattice of L
consisting of elements fixed by G is tri
Graded left modular lattices are supersolvable
We provide a direct proof that a finite graded lattice with a maximal chain
of left modular elements is supersolvable. This result was first established
via a detour through EL-labellings in [McNamara-Thomas] by combining results of
McNamara and Liu. As part of our proof, we show that the maximum graded
quotient of the free product of a chain and a single-element lattice is finite
and distributive.Comment: 7 pages; 2 figures. Version 2: typos and a small error corrected;
diagrams prettier; exposition improved following referee's suggestions;
version to appear in Algebra Universali
Trimness of Closed Intervals in Cambrian Semilattices
In this article, we give a short algebraic proof that all closed intervals in
a -Cambrian semilattice are trim for any Coxeter
group and any Coxeter element . This means that if such an
interval has length , then there exists a maximal chain of length
consisting of left-modular elements, and there are precisely join- and
meet-irreducible elements in this interval. Consequently every graded interval
in is distributive. This problem was open for any
Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my
now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure
A new subgroup lattice characterization of finite solvable groups
We show that if G is a finite group then no chain of modular elements in its
subgroup lattice L(G) is longer than a chief series. Also, we show that if G is
a nonsolvable finite group then every maximal chain in L(G) has length at least
two more than that of the chief length of G, thereby providing a converse of a
result of J. Kohler. Our results enable us to give a new characterization of
finite solvable groups involving only the combinatorics of subgroup lattices.
Namely, a finite group G is solvable if and only if L(G) contains a maximal
chain X and a chain M consisting entirely of modular elements, such that X and
M have the same length.Comment: 15 pages; v2 has minor changes for publication; v3 minor typos fixe
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