| Let the natural numbers n≥3.Tn and OPn be full transformed semigroups and orientation-preserving transformed semigroups on a finite chain[n],respectively.Letα ∈ Tn,if for any 1≤<k≤n.satisfy((?)x ∈[n])x≤k(?)xα ≤k,and(1α,2α,…,kα)is a cyclic,that is,there is at most one natural number i≤k,such as iα>(i+1)α,then α is said to be local k-type orientation-preserving.It is easy to prove that the set of all local k-type orientation-preserving elements in Tn is a semigroup with respect to the multiplica-tion of the transformation,which is called the local orientation-preserving transformation semigroup,dennoted by TOPn(k).Let TOPnk={α∈TOPn(k):x≤k(?)xα≤k},It is proved that TOPnk is a regular subsemigroup of the semigroup TOPn(k),It is called a local regular orientation-preserving transformation semigroup.OSk={α∈TOPn(k):|im(α|=n},SOPn(k)=TOPn(k)\OSk and SOPnk=TOPnk\OSk are said to be local orientation-preserving symmetry group,singular local orientation-preserving transformation semigroup and singular local regular orientation-preserving transformation semigroup,respectively.Let I(n,r)(k)={α∈SOPn(k)|im(α)|≤r},I(n,r)k{α∈SOPnk:|im(α)|≤r}.In this paper,the Green(*)-relation of the local(regular)orientation-preserving trans-formation semigroups is described,the rank of ideals of the local(regular)orientation-preserving transformation semigroups and maximal subsemigroups of the semigroup TOPn(k)are studied.The details are as follows:In capter 1 We give introduction and preliminaryIn capter 2 By studying the elemental characteristics and egg box graph of semigroup TOPn(k),the Green(*)-relation of semigroup TOPn{k)is characterized,and it is further proved that when 2≤k≤n-1,the semigroup TOPn(k)is a non-regular abundant semigroup.In chapter 3 by analyzing the element of the semigroup TOPn(k)with rank r,we describe the characteristics of the elements generated by the regular element,and get that the set generated by the regular element is exactly the semigroup TOPn(k),and then obtain the local(regular)orientation-preserving transformation semigroup and its ideal rank.Finally,the maximal subsemigroups of TOPn(k)are classified based on the results of the research rankIn chapter 4,summary and prospect... |
