Some notes on semi-ordered sets which form positive Cones of semi-ordered vector spaces

摘要

Let R be a semi-ordered vector space. We put R^＋=｛ｘ：ｘ∈R，ｘ≧0｝　　R^＋ is a semi-ordered set with the least element 0 and satisfies the following conditions:　　　　I.(1)for every a，b∈R^＋ we have a＋b∈R^＋，　　　　　　(2)(a＋b)＋c=a＋(b＋c)，　　　　　　(3)a＋b=b＋ａ，　　　　　　(4)a＋0＝a，　　　　　　(5)for every a，b∈R^＋ we have a≦a＋b，　　　　　　(6)a＋c＝b＋c implies a＝b，　　　　　　(7)if a≦ｂ，then we have uniquely determined ｃ∈R^＋　such that a＋c＝b，　　　　II.(1)for any real number α≧O and a∈R^＋ we haveαa∈R^＋，　　　　　　(2)α(βa)＝(αβ)a，　　　　　　(3)α(a＋b)＝αa＋αb，　　　　　　(4)(α＋β)a＝αa＋βa，　　　　　　(5)1a＝a，　　Generally let x be a semi-ordered set and we assume that x satisfies the previous conditions l. (1)-(7)and II.(1)-(5). In this paper，such a semi-ordered set x is styled as “a semi-ordered set with the least element and non-negative real domain of operators". In this paper we discuss the existence of a semi-ordered vector space R which R^＋＝X and some properties of x.