55 research outputs found
Notes on the description of join-distributive lattices by permutations
Let L be a join-distributive lattice with length n and width(Ji L) \leq k.
There are two ways to describe L by k-1 permutations acting on an n-element
set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a
recent lattice theoretical way of the second author. We prove that these two
approaches are equivalent. Also, we characterize join-distributive lattices by
trajectories.Comment: 8 pages, 1 figur
On Semigroups with Lower Semimodular Lattice of Subsemigroups
The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green\u27s relation J is trivial have this property, a description in such terms is natural. In the case of periodic semigroups—a case that turns out to include all eventually regular semigroups—the characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely simple, semigroup with this property. Any such semigroup must at least be idempotent-free and D-trivial
Lower Semimodular Inverse Semigroups, II
The authors’ description of the inverse semigroups S for which the lattice ℒℱ(S) of full inverse subsemigroups is lower semimodular is used to describe those for which (a) the lattice ℒ(S) of all inverse subsemigroups or (b) the lattice �o(S) of convex inverse subsemigroups has that property. In each case, we show that this occurs if and only if the entire lattice is a subdirect product of ℒℱ(S) with ℒ(E S ), or �o(E S ), respectively, where E S is the semilattice of idempotents of S; a simple necessary and sufficient condition is found for each decomposition. For a semilattice E, ℒ(E) is in fact always lower semimodular, and �o(E) is lower semimodular if and only if E is a tree. The conjunction of these results leads to quite a divergence between the ultimate descriptions in the two cases, ℒ(S) and �o(S), with the latter being substantially richer
Congruences in slim, planar, semimodular lattices: The Swing Lemma
In an earlier paper, to describe how a congruence spreads from a prime
interval to another in a finite lattice, I introduced the concept of
prime-perspectivity and its transitive extension, prime-projectivity and proved
the Prime-projectivity Lemma. In this paper, I specialize the
Prime-projectivity Lemma to slim, planar, semimodular lattices to obtain the
Swing Lemma, a very powerful description of the congruence generated by a prime
interval in this special class of lattices
The Lattice of Cyclic Flats of a Matroid
A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats
of a matroid form a lattice under inclusion. We study these lattices and
explore matroids from the perspective of cyclic flats. In particular, we show
that every lattice is isomorphic to the lattice of cyclic flats of a matroid.
We give a necessary and sufficient condition for a lattice Z of sets and a
function r on Z to be the lattice of cyclic flats of a matroid and the
restriction of the corresponding rank function to Z. We define cyclic width and
show that this concept gives rise to minor-closed, dual-closed classes of
matroids, two of which contain only transversal matroids.Comment: 15 pages, 1 figure. The new version addresses earlier work by Julie
Sims that the authors learned of after submitting the first versio
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