have both a least upper bound, called the join or supremum, denoted by
, and a greatest lower bound, called the meet or infimum, denoted by
The following definitions apply to posets in general, not just lattices, except where otherwise stated.
Like a geometric lattice, a matroid is endowed with a rank function, but that function maps a set of matroid elements to a number rather than taking a lattice element as its argument.
The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and it must be submodular, meaning that it obeys an inequality similar to the one for semimodular ranked lattices: for sets X and Y of matroid elements.
The maximal sets of a given rank are called flats.
This rank function is necessarily monotonic and submodular, so it defines a matroid.
This matroid is necessarily simple, meaning that every two-element set has rank two.
[4] There are two different natural notions of duality for a geometric lattice
Cheung (1974) defines the adjoint of a geometric lattice