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


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