Gaussian, Strong and Transversal Greedoids

Abstract

Various types of greedoids have been studied in relation to the Greedy Algorithm and therefore it is not surprising that these structures are inter-linked in various ways. However, it is quite surprising that the Gauss greedoids and the strong greedoids, approached from quite different points of view, turn out to be precisely the same. Although this can be seen very indirectly by combining the two sets of separate results, the main purpose of this short paper is to prove directly that the two structures are identical, thus giving a simple axiomatic characterisation of the Gauss greedoids. We also note that the transversal greedoids provide an interesting class of examples of such greedoids. c 1999 Academic Press THE GAUSS GREEDOIDSRecall that a greedoid (E, E) consists of a finite set E and a collection of subsets E of E which satisfy:In both cases, the members of E are called the independent sets, the maximal independent sets are the bases, and the common cardinality of the bases is the rank of the structure. More generally, the rank of a set is the cardinality of its maximal independent subsets, and a flat is a maximal set of any given rank. All the standard terminology and results concerning matroids can be found in [6] and the corresponding results for greedoids in [5].We now consider some greedoids motivated by the work of Goecke [3, 4] (and recalled in [5]).DEFINITION. An inclusion chain = (E 1 , E 2 , . . . , E n ) consists of a family of matroids on a ground set E, where each E i ⊂ E i+1 and where the rank of each E i equals i. Given such a define the associated Gaussian structure G( ) as the collection of subsets of E consisting of the empty set ∅ together with all sets of the form {x 1 , x 2 , . . . , x k } such that {x 1 , x 2 , . . . , x i } ∈ E i for 1 i k.It is straightforward to check that a Gaussian structures is a greedoid. Note also that each of its non-empty independent sets is a basis of one of the matroids in the inclusion chain, but that not all such bases are necessarily independent in the Gaussian structure. Now in order to consider a strengthened form of an inclusion chain based on Goecke's work using 'strong maps' we need the following result: KEY LEMMA. Let (E, E 1 ) be a matroid of rank one less than the rank of the matroid (E, E 2 ). Then the following properties are equivalent:(1) Every flat of E 1 is also a flat of E 2 .(2) E 1 ⊂ E 2 and, given bases B 1 and B 2 of (E, E 1 ) and (E, E 2 ), respectively, there is anx ∈ B 2 \ B 1 such that B 1 ∪ {x} ∈ E 2 and B 2 \ {x} ∈ E 1 .

A Greedoid and a Matroid Inspired by Bhargava's $p$-Orderings

“…, v i−1 have already been constructed). 5 (These projections v i exist because of Proposition 11 6 ; they may be non-unique, but any choice is fine.) Thus, we get k elements v 1 , v 2 , .…”

“…This definition of strong greedoids appears in [6] (where the above axiom (iv) appears as property G( 3) ). Note that axiom (iv) is clearly stronger than axiom (iii).…”

“…Note that axiom (iv) is clearly stronger than axiom (iii). The theorem in Section 2 of [6] says that strong greedoids are the same as Gauss greedoids (one of the classes of greedoids studied in [10]). See [10, Section IX.4] for further properties and characterizations of Gauss greedoids.…”

“…(Here, a triangle means a 3-subset of E; its sides are its 2-subsets; the length of a side {u, v} is d (u, v).) the electronic journal of combinatorics 28(3) (2021), #P3 6.…”

“…See [10, Theorem II.1.1] for a proof of its equivalence to other definitions of a matroid. See also[13] for much more about matroids.the electronic journal of combinatorics 28(3) (2021), #P3 6.…”