49 research outputs found
Characterizing fully principal congruence representable distributive lattices
Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice
is said to be fully principal congruence representable if for every subset
of containing , , and the set of nonzero join-irreducible
elements of , there exists a finite lattice and an isomorphism from the
congruence lattice of onto such that corresponds to the set of
principal congruences of under this isomorphism. Based on earlier results
of G. Gr\"atzer, H. Lakser, and the present author, we prove that a finite
distributive lattice is fully principal congruence representable if and
only if it is planar and it has at most one join-reducible coatom. Furthermore,
even the automorphism group of can arbitrarily be stipulated in this case.
Also, we generalize a recent result of G. Gr\"atzer on principal congruence
representable subsets of a distributive lattice whose top element is
join-irreducible by proving that the automorphism group of the lattice we
construct can be arbitrary.Comment: 20 pages, 8 figure
CD-independent subsets in meet-distributive lattices
A subset of a finite lattice is CD-independent if the meet of any two
incomparable elements of equals 0. In 2009, Cz\'edli, Hartmann and Schmidt
proved that any two maximal CD-independent subsets of a finite distributive
lattice have the same number of elements. In this paper, we prove that if
is a finite meet-distributive lattice, then the size of every CD-independent
subset of is at most the number of atoms of plus the length of . If,
in addition, there is no three-element antichain of meet-irreducible elements,
then we give a recursive description of maximal CD-independent subsets.
Finally, to give an application of CD-independent subsets, we give a new
approach to count islands on a rectangular board.Comment: 14 pages, 4 figure
Varieties of distributive rotational lattices
A rotational lattice is a structure (L;\vee,\wedge, g) where
L=(L;\vee,\wedge) is a lattice and g is a lattice automorphism of finite order.
We describe the subdirectly irreducible distributive rotational lattices. Using
J\'onsson's lemma, this leads to a description of all varieties of distributive
rotational lattices.Comment: 7 page
Embedding convex geometries and a bound on convex dimension
The notion of an abstract convex geometry offers an abstraction of the
standard notion of convexity in a linear space. Kashiwabara, Nakamura and
Okamoto introduce the notion of a generalized convex shelling into
and prove that a convex geometry may always be represented with such a
shelling. We provide a new, shorter proof of their result using a recent
representation theorem of Richter and Rubinstein, and deduce a different upper
bound on the dimension of the shelling.Comment: - Corrected attribution for Lemma 1 and Theorem 2 - Added an example
related to generalized convex shellings of lower-bounded lattices and noted
its relevance to convex dimension. - Added a section on embedding convex
geometries as convex polygons, including a proof that any convex geometry may
be embedded as convex polygons in R^2. - Extended the bibliography. Now 9
page