Some Studies On Transformation Semigroups And Endomorphism Monoids Of Graphs

This Ph.D.dissertation consists of six chapters.We study the properties and constructions of the certain order-preserving transformation semigroups and the endomorphism monoids of graphs.In chapter 2,a class of order-preserving transformation semigroups (?)(X,Y)which preserve Y invariant is explored.We characterize the Green’s-relations and Green’s*-relations on o(X,Y),meanwhile show that (?)(X,Y)is abundant but not a regular semigroup.Then we describe the regular elements of (?)(X,Y)and determine when (?)(X,Y)is a regular semigroup.Furthermore,we compute the cardinalities of E((?)(X,Y)),Reg((?)(X,Y))and (?)(X,Y),respectively.In chapter 3,we assign (?)(X,Y)with so-called natural partial order.Under this partial order,we determine when two elements of (?)(X,Y)are related,find the compatible elements,the maximal elements,the minimal elements,and the greatest lower bound of two elements.In chapter 4,we determine the End-regular generalized lexicographic products of bipartite graphs G[Bx],We show the condition that when G[Bx]is End-regular for G = K1,n.For d(G)≥3,we give the sufficiency and necessity for End-regular of G[Bx]via the definitions of the distance between two K3 subgraphs and two K4 subgraphs,when G[Bx]does and does not contain of K4 subgraphs.In chapter 5,we study the connected 4-regular Cayley graph г by representing them as pseudo-cartesian product of two cycles.We characterize the automorphism groups and endomorphism monoids on г，and prove that endomorphism monoids of pseudo-cartesian product of two cycles are isomorphism if and only if graphs are isomorphism.We compute the cardinalities of End(г)and determine the condition under which End(г)is regular,completely regular and orthodox,respectively.Fur-thermore,we characterize the endomorphism spectrum and endomorphism type of End-regular г.In chapter 6,we characterize the endomorphism monoids of 8-graphs.We show the Greenes relations and regular D-classes of endomorphism monoids of 8-graphs.