Polystroma

Polystromata are combinatorial objects introduced by Branko Grünbaum. They are considered a precursor to the modern concept of abstract polytopes.

Definition

A polystroma is a partially ordered set with a single minimum element.

A polystroma may also be defined to have a maximum element. Since there is a simple bijection between posets without a maximum element and those with (add a new element greater than all existing elements) this definition is functionally equivalent. As long as the definition used is consistent it makes no difference to most relevant properties.

Regular polystromata

While the definition of of polystroma is very general, the primary subjects of interest regular polystromata have considerably more structure.

Definition

A flag of a polystroma 𝓟 is a totally ordered subset of 𝓟 which is not a proper subset of any other totally ordered subset of 𝓟.

The automorphism group of a polystroma 𝓟 is the group of order preserving bijections on the elements of 𝓟.

A polystroma is regular iff its automorphism group acts transitively on its flags.

Properties

Regularity imposes some additional properties that make polystroma more like abstract polytopes.

Since the automorphism group is transitive on each flag, the order type of each flag is the same. This means that elements can be assigned a rank, as an element of that order type, and thus this makes the regular polystromata a subset of regular incidence geometries.

Comparison to abstract polytopes

The definition of polystromata is much weaker than that of abstract polytopes. Polystromata are not required to be connected or to meet the diamond condition among other considerations. This specificity of abstract polytopes has made them richer objects of study.

Bibliography

• Grünbaum, Branko (1976). "Regularity of Graphs, Complexes and Designs" (PDF). Problèms Combinatoire et Théorie Theorie des Graphes (260): 191–197.