Research On Special Elements And Maximal Substructures Of Finite Transformation Semigroups

Transformation semigroups are not only important research contents of semigroup algebra theory' but also widely used in theoretical computer science,formal language theory,cryptography and other disciplines.This paper combines the knowledge of“semigroup algebra theory" and“transformed semigroup theory",and studies the substructures of finite transform semigroups by means of maximum method and special element method.The specific work is as follows:(1)The structure and classification of subsemigroups of common transformation semigroups are studied.A new concept of maximality is introduced into common transformation semigroups.The study of subsemigroups of finite transformation semigroups with some maximal properties is expanded,and the structure theory of finite semigroups and finite transformation semigroups is enriched.The structure and complete classification of the maximal subsemibands of the ideals of the partial transformation semigroup Pn,order-preserving partial transformation semigroup POn,orientation-preserving transformation semigroup OPn and orientation-preserving partial transformation semigroup POPn are studied,and their complete classification are obtained.The structure and complete classification of the maximal regular subsemigroups of the ideal of the orientation-preserving partial transformation semigroup POPn,and orientation-preserving partial transformation semigroup POPn are characterized.Considering the structure of the local maximal subsemibands of the ideal of full transformation semigroup Tn,its complete classification is obtained.The structure and complete classification of the maximal subsemigroups and maximal idempotent generating subsemigroups of the ideals of the transformation semigroup PO?n are determined.(2)The related concepts in semigroup theory are introduced into transformation semigroups,and the related contents of finite transformation semigroups are studied.We introduce the concept of index and period of elements in semigroup theory into transformation semigroups and propose the concept of the(p,q)-potent rank of transformation semigroups.We introduce m-potent rank in directionally orientation-preserving partial transformation semigroups POPn and strictly orientation-preserving partial transformation semigroups POPn,the m-potent rank of their ideals are obtained.The concept of(p,q)-potent rank are introduced into the full transformation semigroup Tn,the partial transformation semigroup Pn,and the strict partial transformation semigroup SPn.The(p,q)-potent rank of their ideals are obtained,which generalizes the results of the existing literature.The sufficient and necessary conditions for the minimal generating set of the ideals of the orientation-preserving transformation semigroup are studied.Secondly,the concept of the nilpotent rank of the principal factor is introduced on the ideals of the orientation-preserving transformation semigroup,and its nilpotent rank of the principal factor is obtained.The structure of the subsemigroup of the order-preserving transformation semigroup On generated by the idempotent of rank n-1 is deeply studied,and their rank and idempotent rank are obtained,which generalizes the work of Howie and Higgins.(3)We construct new transformation semigroups and study the related contents of new transformation semigroups.We mainly construct new transformation semigroups with additional conditions(such as A-descending order condition)for common transformation semigroups and study regularity and relation rank of new transformation semigroups.For partial order-preserving transformation semigroups POn with additional A-descending order condition,the rank and idempotent rank of the new transformation semigroup POn(A)are studied.A new transformation semigroup Sn(A)is constructed by adding A-descending order condition to semigroup Singn.The Green star relation and regularity of semigroup Sn(A)are studied,and their ranks and idempotent ranks are determined.It extends the work of Howie and Gomes.