On Permutation Geometries

The lattice of flats of a matroid or combinatorial geometry can be regarded as a sublattice (with rank function) of the lattice of subsets of a set, having the property that, given an element of rank r of the lattice and a point outside it, a unique element of rank r+1 covers both. This paper develops a theory of permutation geometries, which are similarly defined objects in the semilattice of subpefmiitations (partial 1-1 mappings) on a set. As well as providing this basic analogy, raatroid theory is related in other ways to our theory of permutation geometries. In the semilattice of all subpermutations, the interval consisting of all elements below a given bound is a lattice isomorphic to the lattice of all subsets of a set. In a permutation geometry, any such lower interval is the lattice of flats of a matroid. The set of subpermutations, as well as being a semilattice, has additional structure: it is an inverse semigroup, where maximal elements (the permutations) form a group. The most interesting permutation geometries, and those to which we give most attention, are the ones whose maximal elements form a subgroup of the symmetric group. (We call a group which arises in this way a geometric group.) Our results include the determination of doubly transitive geometric groups, and some examples which are not doubly transitive. Connections also exist with the theories of permutation cliques, sharp permutation groups, and Jordan groups. The authors are grateful to C. Landauer for helpful remarks.