Majors of geometric strong maps

Most of the constructions in the theory of combinatorial geometries take place in the category of pregeometries and strong maps. In this present paper, we study these constructions and the structure of pregeometries by factoring strong maps into elementary maps, after the work of Dowling and Kelly. Using the modular cuts of the factorization to determine certain single-element extensions, we associate each factorization @F to a unique labeled pregeometry, called the major of @F. Every major of a factorization of a strong map f:H->G has H and G as distinguished minors: H as a subgeometry; G as a contraction. A partial order is defined on the set of all factorizations of f, , and a compatible partial order is defined on the set of all majors of f, . The map @F : -> is shown to be an increasing surjection, while the map Y : -> is shown to be an increasing injection. The composition @F @? Y is shown to be the identity on , while Y @? @F is a decreasing map on , defining a co-closure on majors. The partial order on , although seemingly more restrictive than necessary, is shown to be necessary to reflect the order on factorizations. Special consideration is given the zero element of , called the Higgs factorization of f, which is constructed using the lift construction of Higgs. We explore the connection between the majors of this theory and a particular major constructed by Higgs, and show that this Higgs major is the major of the Higgs factorization. As the unique smallest major of any strong map, the Higgs major thus defines a unary operator on pregeometries: Y(G) is the Higgs major of f : ->G.