Jordan-Dedekind space

Let $ {\mathcal C} $ be a closure space on a set $ S $. The elements of $ {\mathcal C} $, partially ordered by set-inclusion, form a complete atomic lattice [a3] (cf. also Atom). For any subset $ X $ of $ S $, let $ \langle X \rangle $ denote the closure of $ X $. A chain in a closed set $ A $ is a totally ordered set of closed subsets of $ A $. The rank $ r ( X ) $ of a set $ X $ is

$$ \max \left \{ {\left | M \right | } : {M \textrm{ a chain of } \left \langle X \right \rangle } \right \} - 1 . $$

A Jordan–Dedekind space is a closure space of finite rank satisfying the Jordan–Dedekind chain condition (see Jordan–Dedekind lattice).

Characterizations of Jordan–Dedekind spaces in terms of an exchange property and in terms of independence were given by L.M. Batten in [a1] and [a2]. In particular, let $ {\mathcal C} $ be a closure space. $ {\mathcal C} $ is said to have the weak exchange property if for all elements $ y $ of $ S $ and subsets $ X $ of $ S $,

$$ r ( \left \langle {X \cup \{ y \} } \right \rangle ) = 1 + r ( \left \langle X \right \rangle ) . $$

The following theorem holds: In any closure space of finite rank, the weak exchange property is equivalent to the Jordan–Dedekind chain condition (cf. Jordan–Dedekind property).

The notion of an independent set is recursively defined: $ X $ is independent if $ X = \emptyset $ or a singleton; $ X $ is independent if for some $ x \in X $, $ X \setminus \{ x \} $ is independent and $ x \notin \langle {X \setminus \{ x \} } \rangle $. The set $ X $ is $ m $- independent if for all $ x \in X $, $ x \notin \langle {X \setminus \{ x \} } \rangle $.

The following theorem holds: For a Jordan–Dedekind space $ {\mathcal C} $ the following assertions are equivalent: 1) $ {\mathcal C} $ is a matroid [a4]; and 2) $ m $- independence and independence are the same.

References

[a1]	L.M. Batten, "A rank-associated notion of independence" , Finite Geometries , M. Dekker (1983)
[a2]	L.M. Batten, "Jordan–Dedekind spaces" Quart. J. Math. Oxford , 35 (1984) pp. 373–381
[a3]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , Amer. Math. Soc. (1967) (Edition: Third)
[a4]	D.J.A. Welsh, "Matroid theory" , Acad. Press (1976)