In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism ...In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism monoid, the set of all half-strong endomorphisms, the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X, respectively. The conditions under which hEnd（X） forms a submonoid of End（X） are given. It is shown that 1End（X） = qEnd（X） for any split graph X. The conditions under which 1End（X） （resp. qEnd（X）） forms a submonoid of End（X） are also given. In particular, if hEnd（X） forms a monoid, then 1End（X） （resp. qEnd（X）） forms a monoid too.展开更多
For a prime p,let E_（p,p^m）={（a p^（m-1） b d）｜a,b,c∈Z_p,d∈Z_（p^m）}. We first establish a ring isomorphism from Z_（p,p^m） onto E_（p,p^m）. Then we provide a way to compute-d and d^（-1） by using arithmeti...For a prime p,let E_（p,p^m）={（a p^（m-1） b d）｜a,b,c∈Z_p,d∈Z_（p^m）}. We first establish a ring isomorphism from Z_（p,p^m） onto E_（p,p^m）. Then we provide a way to compute-d and d^（-1） by using arithmetic in Z_p and Z_（p^m）, and characterize the invertible elements of E_（p,p^m）. Moreover, we introduce the minimal polynomial for each element in E_（p,p^m） and give its applications.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos. 10571077,10971086)
文摘In this paper, the half-strong, the locally strong and the quasi-strong endomorphisms of a split graph are investigated. Let X be a split graph and let End（X）, hEnd（X）, 1End（X） and qEnd（X） be the endomorphism monoid, the set of all half-strong endomorphisms, the set of all locally strong endomorphisms and the set of all quasi-strong endomorphisms of X, respectively. The conditions under which hEnd（X） forms a submonoid of End（X） are given. It is shown that 1End（X） = qEnd（X） for any split graph X. The conditions under which 1End（X） （resp. qEnd（X）） forms a submonoid of End（X） are also given. In particular, if hEnd（X） forms a monoid, then 1End（X） （resp. qEnd（X）） forms a monoid too.
基金Supported by the Research Project of Hubei Polytechnic University(17xjz03A)
文摘For a prime p,let E_（p,p^m）={（a p^（m-1） b d）｜a,b,c∈Z_p,d∈Z_（p^m）}. We first establish a ring isomorphism from Z_（p,p^m） onto E_（p,p^m）. Then we provide a way to compute-d and d^（-1） by using arithmetic in Z_p and Z_（p^m）, and characterize the invertible elements of E_（p,p^m）. Moreover, we introduce the minimal polynomial for each element in E_（p,p^m） and give its applications.
