CAJ | 학술논문

可数无限秩的自由格序群是同构于有理数集上的格序置换群A(Q)的2-可迁子群[1,2],THEOREM6.7).McCleary证明了有限秩的自由格序群有一个Q上的2-可迁表示.McCleary给出自由格序群Fη(1＜η＜N0)在Q上有一个o-2-可迁作用[4].这一想法被推广到格序群的自由积.若G是一个L-群,F是基数至少是| F|的无限生成子上的自由群,则自由积G∪H在一个基数|F|的秩域上有一个o-2-可迁表示.Glass和Gurevich则证明了两个可数L-群在Q上有一个o-2-可迁表示[6].证明若G和H是在有理数集Q上有忠实表示的非平凡可数群,则它们的自由积G ∪ H在Q上有高可迁忠实表示;若G和H是非平凡有限和可数群,且H有一个无限阶元素,则自由积G∪H在自然数集上有高可迁忠实表示.
가수무한질적자유격서군시동구우유리수집상적격서치환군A(Q)적2-가천자군[1,2],THEOREM6.7).McCleary증명료유한질적자유격서군유일개Q상적2-가천표시.McCleary급출자유격서군Fη(1＜η＜N0)재Q상유일개o-2-가천작용[4].저일상법피추엄도격서군적자유적.약G시일개L-군,F시기수지소시| F|적무한생성자상적자유군,칙자유적G∪H재일개기수|F|적질역상유일개o-2-가천표시.Glass화Gurevich칙증명료량개가수L-군재Q상유일개o-2-가천표시[6].증명약G화H시재유리수집Q상유충실표시적비평범가수군,칙타문적자유적G ∪ H재Q상유고가천충실표시;약G화H시비평범유한화가수군,차H유일개무한계원소,칙자유적G∪H재자연수집상유고가천충실표시.
The free lattice-ordered group of a countable infinite rank had been shown to be isomorphic to a doubly transitive subgroup of 1-permutation A (Q) of the rational line[1,2], theorem 6.7). McCleary proved that the free lattice-ordered group of finite ranks possessed a (faithful) doubly transitive representation on Q[3]. McCleary proved that the free lattice-ordered group F,, (1 < η≤N0 ) had been in action of an 0-2-transition on Q[4]. The ideas were extended to free products of 1-groups. If G is an 1-group, and F is a free l-group on an infinite generator set of ca_rdinality at least |F|, then the free product GUH has a faithful 0-2-transitive representation on some ordered fields of cardinality |F|[5]. More recently, Glass and Gurevich showed that the free l-product of two countable l-groups has had a faithful 0-2-transitive representation on Q[6]. Let U be a class of groups,{Gi|∈I} be a family of U. Then U- free product of the family {Gi|i∈I},denoted Ui∈IGi is a group G ∈U together with a family of injective homomorphisms { ai: Gi→G|i∈I} such thati) Ui∈Iai) generates G;ii) IfH∈U and {βi:Gi→H}is a family of homomorphisms, there exists an unique hornomorphismv:G→such that βi =μαi for each i∈I.Our main results are Theorem 1 If G and H are nontrivial countable groups having faithful representations as groups of order-preserving of Q, then the product G∪H has such a representation which in addition is highly order-transitive.Theorem 2 If G and H are nontrivial finite or countable groups, and if H has an element of infinite order, then the product G∪H can be faithfully represented as a highly transitive group on N.We have given that if G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G∪H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then free product G∪H can be faithfully represented as a highly transitive group on N.