The prime stems of rooted circuits of closure spaces and minimum implicational bases
type: Item
instance of: scholarly article
title: The prime stems of rooted circuits of closure spaces and minimum implicational bases
zbMATH Open document ID: 1264.06006
author: Masataka Nakamura
author: Kenji Kashiwabara
published in: The Electronic Journal of Combinatorics
publication date: 2013-06-07T00:00:00Z
full work available at URL: v20i1p22
review text: Summary: A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair \((X,e)\) of a subset \(X\), called a stem, and an element \(e\not\in X\), called a root. We introduce a notion called a `prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore gives rise to an optimal basis. A `critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, `critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. This is shown through an example.
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 52B40
zbMATH DE Number: 6171861
zbMATH Keywords: closure system
zbMATH Keywords: closure operator
zbMATH Keywords: Armstrong's axioms
zbMATH Keywords: pseudo-closed set
zbMATH Keywords: convex geometry
zbMATH Keywords: rooted circuit
zbMATH Keywords: precedence structure
zbMATH Keywords: antimatroid
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6171861