Research On Structural Properties Of Several Transformation Semigroups

As the power and source of studying the semigroup algebraic theory,transfor-mation semigroups are not only an important part of semigroup algebraic theory,but also closely related to many other disciplines such as computer science,cryptography and graph theory.In this thesis,we maily characterize the monoids of endomorphisms of three classes of semigroups of order-preserving or order-reversing transformations of a finite chain,as well as the monoids of endomorphisms of six classes of semi-groups of orientation-preserving or orientation-reversing transformations of a finite chain,meanwhile,the cardinal of each of these nine monoids of endomorphisms are determined.In addition,we also study some algebraic properties,rank properties and combinatorial properties of another four classes of transformation semigroups.In Chapter 1,we introduce the related backgrounds and motivations of our re-search.For the completeness of this thesis,we also recall some definitions and notation that will be used.In Chapter 2,we characterize the monoid of endomorphisms of the semigroupOD_nof all order-preserving or order-reversing full transformations of a finite chain,as well as the monoids of endomorphisms of the semigroupPOD_nof all order-preserving or order-reversing partial transformations and of the semigroupPODI_nof all order-preserving or order-reversing partial permutations of a finite chain.In addition,we compute the cardinal of each of these three monoids of endomorphisms.In Chapter 3,we characterize the monoid of endomorphisms of the semigroupOR_nof all orientation-preserving or orientation-reversing full transformations of a finite chain,as well as the monoid of endomorphisms of the semigroupPOR_nof all orientation-preserving or orientation-reversing partial transformations and the monoid of endomorphisms of the semigroupPORI_nof all orientation-preserving or orientation-reversing partial permutations of a finite chain.Characterizations of the monoids of endomorphisms of the subsemigroupsOP_n,POP_nandPOPI_nof al-l orientation-preserving transformations of the three semigroups aforementioned are also given.In addition,we compute the cardinal of each of these six monoids of endomorphisms.In Chapter 4,we study some structural properties of the semigroupOPE_nof all orientation-preserving and extensive full transformations and of the semigroupPOPEI_nof all orientation-preserving and extensive partial permutations of a finite chain.We characterize Green’s equivalences,regularity,minimal generating sets and rank properties of them.In addition,we also consider the classifications of maximal substructures of these two transformation semigroups,such as the classifications of maximal subsemigroups,maximal idempotent generated subsemigroups,as well as maximal quasi-idempotent generated subsemigroups of them.In Chapter 5,we study some structural properties of the semigroupOPE_n,kof all orientation-preservingk-extensive full transformations of a finite chain.We first consider Green’s relations and regularity ofOPE_n,k.Then we characterize Green’s*–relations ofOPE_n,kand show thatOPE_n,kis an abundant semigroup.Finally,a minimal generating set and a minimal idempotent generating set of the semigroupOPE_n,kare determined,and so the rank and the idempotent rank ofOPE_n,kare established.In Chapter 6,we study some structural properties of the semigroupPOL（X,Y）of all order-preserving partial permutations of a finite chainX with range contained inY.We first characterize Green’s relations and regularity ofPOL（X,Y）,and a sufficient and necessary condition under whichPOL（X,Y）is a regular semigroup is given.Then we describe Green’s*–relationsL^*andR^*onPOL（X,Y）and show thatPOL（X,Y）is left abundant but not right abundant whenY is a proper and nonempty subset ofX.Furthermore,the cardinality of the semigroupPOL（X,Y）is determined,and an isomorphism theorem forPOL（X,Y）is given.Finally,the rank of the semigroupPOL（X,Y）is established.