Partial Orders On Orthodox Semigroups

Algebraic theory of partially ordered semigroups is still one of the most active fields of algebra.The goal of this dissertation is to study partial orders on orthodox semigroups.It mainly achieved in the following aspects:1.We studied amenable partial orders on inverse semigroups.We proved concisely that an amenable partial order on an inverse semigroup is uniquely determined by a McAlister-cone.In particular,we studied the properties and structure of amenable partial orders on Clifford semigroups and obtained a structural method of them.2.We studied amenable partial orders on locally inverse semigroups with inverse transversals.Given S is a locally inverse semigroup with an inverse transversal S°, then there is an order-preserving bijection from the set of all amenable partial orders on S to the set of all McAlister-cones of S°;we showed that an amenable partial order on S is uniquely determined by an amenable partial order on S°and an equivalent characterization of locally inverse semigroup with a Clifford transversal is given.If S is a locally inverse Semigroup with a Clifford transversal,then there also exists an order-preserving bijection between all amenable partial orders on S and all R-cones of S.Also,the result that a regular orthodox semigroup with a Clifford transversal is completely regular is proved.3.We discussed partial orders on right inverse semigroups.With the help of self-conjugate strongly full subsemigroup,we constructed partial orders on right inverse semigroup.We showed that a left amenable partial order on the quotient semigroup of right inverse semigroup(the quotient semigroup is an inverse semigroup)is uniquely determined by a locally maximal cone of right inverse semigroup;extended the amenable partial orders and cones of inverse semigroups to normal orthogroups,and proved that an amenable partial order on a normal orthogroup is uniquely determined by a cone.4.We deliberated semilattice-ordered inverse semigroups.An equivalent characterization of naturally semilattice-ordered inverse semigroups is given;a definite classification of congruence simple semilattice-ordered Clifford semigroups is obtained;in the class of all naturally semilattice-ordered Clifford semigroups,naturally semilattice-ordered zero groups are the only subdirectly irreducible members;we obtained the result that aⅤ-semilatticed left,amenable partially ordered inverse semigroup is an amenable lattice-ordered Clifford semigroup;particularly,we studied amenable latticeordered Clifford semigroups and obtained some interesting results.