Definition
A combinatorial pregeometry (also known as a finitary matroid), is a second-order structure:, where (called the closure map) satisfies the following axioms. For all and :
- is an homomorphism in the category of partial orders (monotone increasing), and dominates (I.e. implies .) and is idempotent.
- Finite character: For each there is some finite with .
- Exchange principle: If, then (and hence by Montonicity and Idempotence in Fact ).
A geometry is a pregeometry where the closure map also satisfies:
- The closure of singletons are singletons and the closure of the empty set is the empty set.
It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are preserved in the framework of abstract geometries.
Let be a pregeometry. We define a topology on by declaring the closed sets to be the fixed points of the closure map (hence by idempotence and monotonicity is the (topological) closure of .) We say for that generates in case . We declare a subset independent if none of its proper subsets generate it.
For, if is independent and generates, we will say that is a base for . Equivalently, a base for is a minimal -generating set, or a maximal independent Subset of .
Read more about this topic: Pregeometry (model Theory)
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