Weak maps of combinatorial geometries

Lucas, Dean

doi:10.1090/S0002-9947-1975-0371693-2

Weak maps of combinatorial geometries
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Trans. Amer. Math. Soc. 206 (1975), 247-279 Request permission

Abstract:

Weak maps of combinatorial geometries are studied, with particular emphasis on rank preserving weak bijections. Equivalent conditions for maps to be reversed under duality are given. It is shown that each simple image (on the same rank) of a binary geometry $G$ is of the form $G/F \oplus F$ for some subgeometry $F$ of $G$. The behavior of invariants under mappings is studied. The Tutte polynomial, Whitney numbers of both kinds, and the Möbius function are shown to behave systematically under rank preserving weak maps. A weak map lattice is presented and, through it, the lattices of elementary images and preimages of a fixed geometry are studied.

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