View source for Jordan-Dedekind space

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Let  $  {\mathcal C} $
be a [[Closure space|closure space]] on a set  $  S $. 
The elements of  $  {\mathcal C} $, 
partially ordered by set-inclusion, form a complete atomic [[Lattice|lattice]] [[#References|[a3]]] (cf. also [[Atom|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|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 [[#References|[a1]]] and [[#References|[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|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|matroid]] [[#References|[a4]]]; and 2)  $  m $-
independence and independence are the same.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.M. Batten, "A rank-associated notion of independence" , ''Finite Geometries'' , M. Dekker  (1983) {{ZBL|0511.05026}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  L.M. Batten, "Jordan–Dedekind spaces"  ''Quart. J. Math. Oxford'' , '''35'''  (1984)  pp. 373–381</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , Amer. Math. Soc.  (1967)  (Edition: Third)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J.A. Welsh, "Matroid theory" , Acad. Press  (1976)</TD></TR>
</table>