The Graham-houghton Graphs Of Full Linear Monoids

It is always a popular research problem to transform algebraic problems into equivalent graph theory problems.Classification problems and infinitesimal family of small order problems are the key research objects in algebraic theory.The Graham-Houghton graph of a semigroup is a bipartite graph with one part indexed by the R-classes of the semigroup,the other part indexed by the L-classes,and edges corresponding precisely to those H-classes that contain idempotents.Graham introduced these graphs to study the idempotent generated subsemigroups of a 0-simple semigroup.Graham’s results were later rediscovered by Houghton who gave them a topological interpretation.In the case that the semigroup is regular and idempotent generated,the connected components of this graph are in natural bijective correspondence with the D-classes of the semigroup.Graham’ s results show that these graphs are important tools for studying idempotent generated semigroups.The matrix semigroup formed by considering the irreversible matrix under multiplication is a regular semigroup,which is a special class of full transformation semigroup.In this paper,we study two special types of Graham-Houghton graphs of full matrix algebras.We illustrate the Graham-Houghton graphs of semigroups of small order matrices over a finite field.We also divide the Green equivalences in some examples of full matrix algebras over a finite field and over a modular n residual ring,and we give the counting formula of idempotents in M2(Fq).At the same time,we illustrate the Graham-Houghton graphs of the full matrix algebraa on the modular n residual ring Z/nZ for the more general n≤10.The two main results in this paper are that the Graham-Houghton graphs corresponding to the D-classes of M2(Fq)are precisely bi-regular graphs with the degree of q,and the Graham-Houghton graphs corresponding to the D-classes containing therepresentation element(?)in M2(Z/nZ)are precisely bi-regular graphs with the degree of n.