The monad is finitary, i.e., and , then there is a finite such that .
(Exchange condition) If , , and , then or . (Cf. matroid)
A geometry is a pregeometry such that and for all .
Let be a vector space, and let be the monad on whose algebras are vector subspaces of . Clearly is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus is a pregeometry.
Similarly, let be a projective space , and let be the monad on whose algebras are projective subspaces. Then is a geometry (the closure of a point is a point). Any pregeometry gives rise to a geometry in a similar way, in the sense that a pregeometry induces a geometry on the image of the function , .
Let be an algebraically closed field; let be the monad on whose algebras are algebraically closed subfields. Then is a pregeometry. That the exchange condition is satisfied is a result due to Steinitz.
Given a pregeometry , a subset is independent if for all , . An independent set said to be a basis for if . All bases of have the same cardinality (?), called the dimension of .