Characterizations of the anti-exchange closure (Q1058534)

Let X be a finite set. A closure operator C on \({\mathcal P}(X)\) is said to be anti-exchange if for each C-closed \(A\in {\mathcal P}(X)\) and any unequal x,y\(\in X\), with x,y\(\not\in A\) if \(x\in C(A\cup \{y\})\) then \(y\not\in C(A\cup \{x\})\). The author studies conditions under which a closure operator on \({\mathcal P}(X)\) is anti-exchange.