43,518 research outputs found
Homology of Distributive Lattices
We outline the theory of sets with distributive operations: multishelves and
multispindles, with examples provided by semi-lattices, lattices and skew
lattices. For every such a structure we define multi-term distributive homology
and show some of its properties. The main result is a complete formula for the
homology of a finite distributive lattice. We also indicate the answer for
unital spindles and conjecture the general formula for semi-lattices and some
skew lattices. Then we propose a generalization of a lattice as a set with a
number of idempotent operations satisfying the absorption law.Comment: 30 pages, 3 tables, 3 figure
Antichain cutsets of strongly connected posets
Rival and Zaguia showed that the antichain cutsets of a finite Boolean
lattice are exactly the level sets. We show that a similar characterization of
antichain cutsets holds for any strongly connected poset of locally finite
height. As a corollary, we get such a characterization for semimodular
lattices, supersolvable lattices, Bruhat orders, locally shellable lattices,
and many more. We also consider a generalization to strongly connected
hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio
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
Exact entropy of dimer coverings for a class of lattices in three or more dimensions
We construct a class of lattices in three and higher dimensions for which the
number of dimer coverings can be determined exactly using elementary arguments.
These lattices are a generalization of the two-dimensional kagome lattice, and
the method also works for graphs without translational symmetry. The partition
function for dimer coverings on these lattices can be determined also for a
class of assignments of different activities to different edges.Comment: 4 pages, 2 figures; added results on partition function when
different edges have different weights; modified abstract; added reference
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