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On poset matroids
Author
Ng, Hwee Ming
Supervisor
Zhao, Dongsheng
Abstract
The theory of poset matroids is the extension of the theory of matroids developed by replacing the underlying set of a matroid by a partially ordered set. Consequently, the notion of subset of a set is replaced by that of filter, or dually, order ideal of a poset. Simply to say, the modern theory of poset matroids is a generalization of the classical theory of matroids.
All features of matroids relating to independence, bases, ranks and circuits can be inserted within the notion of poset matroid. However, not all properties of matroids can be translated in a simple way into this new language. In this project, we shall only consider these features in detail.
For poset matroids, we will be using two languages, namely the language of partially ordered sets, and the language of distributive lattices. By a fundamental theorem of G. Birkhoff, every finite distributive lattice is isomorphic to the lattice of all filters of a finite partially ordered set. Conversely, every finite partially ordered set is isomorphic CO the partially ordered set of the meet-irreducible e1ement.s of a distributive lattice. By virtue of these isomorphisms, we may use the language of posets and the language of distributive lattice interchangeably. The translation of poset matroids into the language of distributive lattices leads to the definition of a combinatorial scheme.
All features of matroids relating to independence, bases, ranks and circuits can be inserted within the notion of poset matroid. However, not all properties of matroids can be translated in a simple way into this new language. In this project, we shall only consider these features in detail.
For poset matroids, we will be using two languages, namely the language of partially ordered sets, and the language of distributive lattices. By a fundamental theorem of G. Birkhoff, every finite distributive lattice is isomorphic to the lattice of all filters of a finite partially ordered set. Conversely, every finite partially ordered set is isomorphic CO the partially ordered set of the meet-irreducible e1ement.s of a distributive lattice. By virtue of these isomorphisms, we may use the language of posets and the language of distributive lattice interchangeably. The translation of poset matroids into the language of distributive lattices leads to the definition of a combinatorial scheme.
Date Issued
2004
Call Number
QA166.6 Ng
Date Submitted
2004