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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 .

In this note the RHC theorems are stated and proved in the setting of a convexity space, a convexity space being a generahzation of a real linear space. For a survey article on the RHC theorems, see DANZER, GR~BAUM, KLEE [1]. In order to make the note self-contained, two introductory sections have been included and much of the material in these sections is due to PEENOWITZ [2; 3]. Convexity spaces have been used by both PRENOWITZ and the authors for generalizing some of the theory of linear spaces. Convexity spacesA more detailed account of the theorems stated in the first two sections of this note, together with their proofs, is given in PRENOWITZ [2]. A convexity space, as defined here, is a modification of PRENOWITZ'S join-system. Throughout, we use X to denote a non-empty set, small letters to denote elements of X, and capitals to denote subsets of X. We find it convenient not to distinguish formally between an element of X and the singleton subset which it defines, and we use notation A ~ B to mean that A n B # r Consider a pair (X, .), where 9 is a function which associates with each ordered pair of elements a, b of X a subset of X called the product of a and b, denoted by a 9 b or simply ab. The most important example of such a pair occurs when X is a real vector space and ab = (2a-~ #b: 2 > 0,/~ > 0, 2-~ # = 1).The product operation 9 enables us to define an inverse operation as follows: if a, b are elements of X, then a/b is the subset of X consisting

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