On The Cross-section Of Semigroup

This thesis consists of four chapters. We shall investigate regular semi-groups with inverse transversals, abundant semigroups with quasi-ideal adequate transver-sals and left adequate semigroups with adequate transversals.In chapter 1, we introduce another new equivalences, 0-Green's relation on semigroups.By appling the relation, we define (left, right) 0-ideals and 0-regular semigroups. Weobtain some properties of 0-Green's relations. We discuss the relations among 0-Green'srelation, Green's relation and *-Green's relation. We make good preparations for chapter2 about inverse transversals.In chapter 2, we study semigroups with inverse transversals and semigroups withorthodox transversals. Suppose that regular semigroup is an enlargement of subsemigroup,by using 0-Green's relations, we give the characterizations of inverse transversals. Someproperties of semigroups with inverse transversals are discussed. We obtain that I is asemilattice of left zero semigroups {L_a;a∈ E(S~0)}, ∧ is a semilattice of right zero seigroups{R_a; a∈E(S~0)}. In section 2.4, we prove that I(S, S~*)/L(?)I(S, S~0)/L. In section 2.5,wegive a characterization of orthodox transversals. We show that if S~o is E~o-full semigroups,then S~o is an orthodox transversal if and only if S~o is an inverse transversals.In chapter 3, we study abundant semigroups with quasi-ideal adequate transversals.The decomposition form of the product of two elements is given in abundant semigroupswith quasi-ideal adequate transversals. An example is given to illustrate that the sets Iand ∧ need not be subbands even if S~o is a quasi-ideal adequate transversal. We give theconditions of equivalence that I and ∧ are subbands. In section 3.3, we show that if anabundant semigroup S has quasi-ideal adequate transversals, if all I and ∧ are subbandsof S, then all quasi-ideal adequate transversals of S are isomorphic. In section 3.4, weprove that if S~* and S~0 are quasi-ideal adequate transversals of abundant semigroup S, ifevery element of E(S~*S~0) commute with all elements of E(S), then S~*S~0 is a quasi-idealadequate transversal of S. In section 3.5, we discuss some properties of abundant semi-groups with quasi-ideal adequate transversals. We show that if I and A are subbands,then I is a semilattice of left zero semigroups {L:; aEE(S')}, A is a semilattice of rightzero semigroups {R:; aEE(S')}, and I and A have a common semilattice transversalE(S'). We study the character of the subsendgroup (E(S)} generated by the idempo-tents of S, an abundant sendgroup with quasi-ideal adequate transversals. An equivalentcondition of multiplicative adequate transversals of abundant semigroups is given.ln chapter 4, we introduce the concept of left adequate semigroups with adequatetransversals. By using an adequate semigroup So and a semilattice I of left zero semi-groups, we define a product set I#S', and show that I#S' is a left adequate semigroupwith an adequate transversal isomorphic with So.