FINITE TOPOLOGIES AND DIGRAPHS

Green's Symmetries in Finite Digraphs

The semigroup DV of digraphs on a set V of n labeled vertices is defined. It is shown that DV is faithfully represented by the semigroup Bn of n × n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in DV follow directly from the Green’s classifications already established for Bn. The new results found from this are: (i) L, R, and H equivalent digraphs...

Digraphs of finite linear transformations

If T is a function from a finite set V to itself, we form the digraph D of T as follows. It has V for its vertex set, and there is an arc from x to T (x) for each x E V. Here we answer the following question. Given a finite field F, which digraphs arise as digraphs of a linear transformation from some finite dimensional vector space over F to itself?

Locally-finite connected-homogeneous digraphs

The classification of finite and locally finite connected-homogeneous digraphs

We classify the finite connected-homogeneous digraphs, as well as the infinite locally finite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.

Uniform Symbolic Topologies and Finite Extensions

We study the behavior of rings with uniform symbolic topologies with respect to finite extensions.