Transversal Lattices

Abstract

A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats are isomorphic to it are transversal. We investigate some sufficient conditions for a lattice to be a Tr-lattice; a corollary is that distributive lattices of dimension at most two are Tr-lattices. We give a necessary condition: each element in a Tr-lattice has at most two covers. We also give constructions that produce new Tr-lattices from known Tr-lattices.

Figures (5)

• Figure 1: The lattice of cyclic flats of a matroid $M$ and that of $M/x$.
• Figure 2: The lattices Acketa considered.
• Figure 3: (a): A generic lattice like $L_8$. (b): A lattice $L_I$ obtained from an ideal in a product of two three-element chains.
• Figure 4: Three nonplanar Tr-lattices; only $D^d$ is MI-orderable.
• Figure 5: (a) The lattice $L_1*L_2$ where $L_1$ and $L_2$ are Boolean lattices on two elements. (b) A lexicographic sum; the indexing lattice is a Boolean lattice on two elements.

Theorems & Definitions (29)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Corollary 3.2
• Definition 3.3
• Theorem 3.4
• Corollary 3.5
• Lemma 3.6
• proof
• Lemma 3.7