## Exchange properties and basis properties for closure operators.(English)Zbl 0711.08003

A closure operator $${\mathcal C}$$ on a set A is called algebraic if, for any $$X\subseteq A$$ and $$a\in {\mathcal C}(X)$$, $$a\in {\mathcal C}(X')$$ for some finite $$X'\subseteq X$$. Furthermore, $${\mathcal C}$$ is said to have the weak exchange property if, for any X,Y$$\subseteq A$$ and $$x\in A$$, $${\mathcal C}(Y)={\mathcal C}(X\cup \{x\})$$ implies $$x\in {\mathcal C}(X\cup \{y\})$$ for some $$y\in Y$$. Let D be a $${\mathcal C}$$-closed subset of A, then $$X\subseteq A$$ is called D-independent if $$x\not\in {\mathcal C}(D\cup X-\{x\})$$, for all $$x\in X$$. $${\mathcal C}$$ is said to have the strong basis property if any D- independent subsets with the same D-closure have the same cardinality. With these definitions, the author shows that an algebraic closure operator on A has the weak exchange property if and only if it has the strong basis property.
Reviewer: G.Eigenthaler

### MSC:

 08A30 Subalgebras, congruence relations 08A05 Structure theory of algebraic structures 20M05 Free semigroups, generators and relations, word problems
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