Consequence Operator - Closure Operators in Algebra

Closure Operators in Algebra

Finitary closure operators play a relatively prominent role in universal algebra, and in this context they are traditionally called algebraic closure operators. Every subset of an algebra generates a subalgebra: the smallest subalgebra containing the set. This gives rise to a finitary closure operator.

Perhaps the best known example for this is the function that associates to every subset of a given vector space its linear span. Similarly, the function that associates to every subset of a given group the subgroup generated by it, and similarly for fields and all other types of algebraic structures.

The linear span in a vector space and the similar algebraic closure in a field both satisfy the exchange property: If x is in the closure of the union of A and {y} but not in the closure of A, then y is in the closure of the union of A and {x}. A finitary closure operator with this property is called a matroid. The dimension of a vector space, or the transcendence degree of a field (over its prime field) is exactly the rank of the corresponding matroid.

The function that maps every subset of a given field to its algebraic closure is also a finitary closure operator, and in general it is different from the operator mentioned before. Finitary closure operators that generalize these two operators are studied in model theory as dcl (for definable closure) and acl (for algebraic closure).

The convex hull in n-dimensional Euclidean space is another example of a finitary closure operator. It satisfies the anti-exchange property: If x is not contained in the union of A and {y}, but in its closure, then y is not contained in the closure of the union of A and {x}. Finitary closure operators with this property give rise to antimatroids.