ORDERED GROUPS
V. M. Kopytov UDC 512.545
In the paper "Ordered groups" there is given a survey of papers on ordered groups, reviewed in RZh Mathematika in 1975-1980.
Starting in 1963 there occurred a qualitative jump in the theory of ordered groups, evoked by the inten- sive investigation of linerarly ordered (l. o. ) groups and the development of the theory of groups of automor- phisms of I.o. sets. As a result many sections of the theory of lattice ordered groups (l-groups) acquired an orderly and organized form, and profound classificationalresults were obtained in them. A natural con- sequence of this was the appearance in the recent past of several monographs on the theory of ordered groups, in particular, the books of Kokorin and Kopytov [26] (English translation [154]), Mura and Rhemtulla [182], Bigard, Keimel, and Wolfenstein [78]. The present survey is written on the materials of the Ref. Zh. "Matematika," mainly for the years 1975-1980, and reflects practicallyall directions of development of the theory of ordered groups with some accent on linearly and lattice ordered groups, which is explained by the intensivity and diversity of the inves- tigations in these domains. In the survey there are included some results reviewed in the years 1970-1974 and not reflected in previous surveys of the collection "Algebra. Topology. Geometry" of the annual 'Itogi Nauki i Tekhniki" [12, 13] or the book [26]. i. Linearly Ordered Groups Investigations on 1. o. groups were carried out mainly in the directions designated at the end of the six- ties and formulated in [26]. The main progress in these directions was made for solvable groups. Tests for Orderability and Description of Linear Orders on Groups. An element x ~ e of the group G is called generalized periodic if there exist gl ..... gn~G such that x- g1-1xgl ... gnlXgn ~ G. Bludov [4] constructed an example of an unorderable group without generalized periodic elements. An example of a solvable group with this property was constructed by Mura and Rhemtulla [179]. They also proved that a solvable group of finite rank without generalized periodic elements has linear orders. Kopytovfound in [29] a test for order- ability of solvable groups of finite rank in terms of irreducible integral polynomials. Rhemtulla [204] proved that if a group can be approximated by finite p-groups for an infinite set of primes p, then it admits linear orders. This result was improved by Kopytov and Medvedev [33]. Namely, such a group admits a linear or- der in which the system of its convex subgroups is central. The latter result allowed them to get by one method a series of facts known earlier about groups, approximable by finite p-groups. Bludov [6] gave a new example of a group admitting a unique linear order. Kopytov[29] proved that if the number of ordered solvable groups is finite, then it is a multiple of 4. For any k > 1 there exists a sol- vable group, admitting precisely 4k linear orders. Pilz [191] characterized all linear orders of the direct pro}luct of linearly ordered groups. Systems of Convex Subgroups in L. O. Groups. NIedvedev studied in [38] orderable groups which have only a finite number of relativelyconvex groups. Each such finitely generated group has nilpotent commutator subgroup, whose quotient-group with respect to the isolator is infinite cyclic, and the rank of such a group is finite. Mura and Rhemtulla [180] proved that each orderable group with a finite number of relativelyconvex subgroups and each of its quotient-groups being a torsion-free group, is hyperorderable. Kop3~ov[29] proved that each solvable [. o. group has a proper normal convex subgroup. He constructs an example of a locally solvable O-simple i.o. group. It is well known that in i. o. groups there holds identicallythe inequality J[x, y]I - ixt. lyJ, and the ine- quality I ix, y] i -< Jxl does not always hold. Kopytovand Medvedev [33] studied I.o. groups for which the latter
Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 19, pp. 3-29, 1981.
0090-4104/83/2306-262757.50 1983 Plenum Publishing Corporation 2627 inequality holds identically. Suchgroups are called rigid I.o. groups. A I.o. group is rigid if and only [f the system of its convex subgroups is central. In the language of normal subgroups conditions are found for the existence on a group of a rigid linear order. Imbedding and Extension. Bludovand Medvedev [9] proved that any orderable metabelian group can be imbedded in a metabelian orderable group in which the equation xn = a is solvable for any element a. Fox in [108, 109] proved that for any Abelian normal subgroup A of the I.o. group G there exists an ordered imbed- ding q~ofthe l o. group G in a i.o. group G* such that the equation xn = q~(a) is solvable in G* for all a eA and integral n. If G is hyperorderable, then G* is also hyperorderable. In the supplement [109] Fox established that if G is contained in the manifold ~ of groups, then G* can also be chosen in ~. Elliott [104] showed that any countable Abelian I.o. group is an inductive limit of a sequence zrl ~-~Izr2q)2 of direct products Zri of cyclic i.o. groups with coordinatewise order, while all the maps ~i are ordered imbeddings. Mura and Rhemtulla proved in [183] that if A is a normal subgroup of a free nonabelian group F, V is a completely characteristic subgroup of A such that A/V is orderable, F/A is right orderable and has an infrain- variant system of subgroups with factors from the manifold of groups generated by A/V, then F/V is order- able. Chehata and Wiegandt [93] constructed a theory of radicals in i.o. groups, analogous to the Kurosh- Amitsur theory for rings. They introduced the concepts of radical and semisimple class of I.o. groups and studied properties of such classes. The class 9~ is radical if and only if 9~ is closed with respect to O-homo- morphisms and transfinite extensions. The class 9~ is semisimple if~is closed with respect to taking normal convex subgroups and transfinite coextensions. There exist radical classes which are simultaneously semi- simple. Properties of L. O. Sets of L. O. Groups. Koppelberg [156] proved that a l.o. group, whose l.o. set is complete and any set of pairwise disjoint nonempty intervals is countable, is separable. Ermolov [19] es- tablished that for any countable I.o. group G and any of its normal subgroups H one can find a countable Abelian group G*, a subgroup H* of it, and a one-for-one map q~:Gof G onto G* such that ~(H) = H* and q~ induces a map of G/H onto G*/H*, where the systems of convex subgroups of G and G~ have the same order type. Goffman [121] analyzed the relation between the concepts of order and topological completeness on the class of i.o. groups In [155] there were considered elementary properties of Abelian I.o. groups, and the decidabilityof the theories of certain concrete classes of Abelian I.o. groups was proved. Right Ordered Groups. Mura and Rhemtulla [181] considered the following subclasses of the class RO of right orderable groups: Cl consists of groups whose system of convex subgroups is solvable for any right order; C2 consists of groups any subgroup of which is a C1-groupi C3 consists of groups G such that for any two elements x, y one can find u, v from the subgroup generated by the elements x, y, such that ux = vy. Then ROD CI D C2 D C3nRO, while all the inclusions are strict. Any locally solvable C3-grou p is locally an extension of a nilpotent group by a finitegroup. A finitely generated solvable orderable C3-grou p is nilpotent. Mura [177] constructed an example of a polycyclic right-orderedgroup, whose system of convex sub- groups is not subnormal. Ault [68, 69] proved that for a locally nilpotent right-ordered group the system of convex subgroups is solvable. Rhemtulla [203] established that if the group G is finitely generated, the quo- tient group by its commutator subgroup G' is finite and G' has nilpotent subgroup of finite index, then G does not admit right orders Each partial right order of a torsion-free nilpotent group extends to a linear right order. An analogous result was also obtained by Formanek [107]. Solvable groups for which each partial right order extends to a linear right order were studied by Pierce [189].
2. Lattice Ordered Groups Properties of Lattices of/-Groups. Completely distributive l-groups were studied by Jakubik [135]. For cardinal numbers a, fl the /-group G is called (a, fl)-distributive if for any sets of indices T, S such that ITI~-(~, ISI--fl the equation A Vxt~= ~/ /k x,~(o holds for any sets {xtslt~T, s~S, Xts~G}, forwhichat tEr s~s ~.s r f~r least one of the terms of this equation is defined. It is proved that any ((~2)-distributive/-group is (a, (~)-dis- stributive. Redfild [199, 200] introduced the conceptofArchimedeanelement, i.e., an element a of the l- group G, such that a > e, and for any g E G, e< g < a, one can find a natural number n, for which gn ~: a. The subgroup A(G) of the l-group G, generated by all Archimedean elements of G, is the largest Archimedean con- vex l-subgroup of G. If G is an /-group in which each element exceeds some Archimedean element, then G is completely distributive if and only if it has a basis.
2628 An element s of the /-group G is called singular if s -> e and for any decomposition s = uv, where u >- e, v >_ e, one has uA v = e. Lattice ordered groups in which each positive element exceeds some singular one, are cailed singular. Wolfenstein [222, 223] characterized singular complete, singular Archimedean, and sin- gular hyperarchimedean (i. e., those l-groups each l-homomorphic image of which is Archimedean) t-groups. This characterization is given in terms of representations of 1-groups by continuous functions, defined on a suitable locally compact completely disconnected set with values in the topological completion Z of the 1. o. group Z of integers in the topology induced by the order. Martinez [163] proved that any sequence a 1 > ... > a n > ... > e of elements of the l-group G such that a n _> a2n+l, terminates if and only if G is the union of an in- creasing sequence of l-ideals Ga, where for any (~, G a +l/Ga is generated by its singular elements. If in addition G is hyperarchimedean, then any l. o. l-homomorphic image of G is cyclic. Berman [73] proved that for any /-group G the semigroup of automorphisms f of the lattice G, for which f(x) - x for any xeG, is transitive. Franchello [112] established that the sublattice L of the l-free product of l-groups Ga, generated by the free factors, is the free product (in the category of distributive lattices with distinguished element e) of the lattices Ga. Systems of Subgroups in l--Groups. Jakubikova proved in [146] that the lattice of all /-subgroups of the l-group G is distributive if and only if G is an Abelian l.o. group of rank 1. We recall that the l-group G is called an l-group with subnormal jumps (or an l-group with normal values), if for any g EG, g r e, any jump Cg < Cg of convex l-subgroups of G, defined by an element g, is subnormal, i. e., Cg is an l-ideal in Cg. The quotient group ~/Cg in this case is an Archimedean 1. o. group. Each quo- tient Cg/Cg is called a component of G. If each component of the l-group G is isomorphic with the group of in- tegers, then G is called generalized discrete. Mott [176] proved that any Abelian generalized discrete/-group can be approximated by generalized discrete Abelian l.o. groups. A countable generalized discrete Abelian 1-group is isomorphic with an l-subgroup of the lexicographic product of 1. o. groups of integers. Conrad and Montgomery [100] established that the classes of /-groups with components of rank 1 and generalized discrete l-groups are closed with respect to l-homomorphisms, taking convex /-subgroups and direct /-products. In an Archimedean singular /-group each 1. o. subgroup is cyclic. Jakubik [136] proved that the converse is false, but ifG is a complete/-group and each of its 1.o. subgroups is cyclic, then G is singular. G. Ya. Rotkovich [57] established that in an r-complete l-group all 1. o. subgroups are cyclic if and only if there are no elements a in it such that for any g, e < g < a, the equation x 2 = g is solvable. Decomposition of l-Groups into a Cartesian l-Product. We recall that by the Cartesian l-product l-[t G~ ~Ez of the l-groups Gc~, aeI, is meant the /-group G of all functions f:I-+ U G~ such that f(oz)eG~, with the opera- a@I tions: (f g) (o:)= f (c~)g (o:), (fVg)(cz)=f(cz)Vg(a), (fAg)(o:)=f(a)Ag(a). By the direct /-product l-It G: is meant the l-subgroup of G, consisting of those f ~G such that f(a) ~ e only for a finite number of o~. Jakubik proved in [133] that if the l-group G is the Cartesian /-product of 1. o. groups Ga, a ~I, and not more than one factor is nonarchimedean, then any of its l-subgroups is a Cartesian l-product of 1. o. groups. One says that the l-group G has the splitting property if G is distinguished as an l-direct factor in any l-~roup H, containing G as an /-ideal. Jakubik [131] studied the splitting property for Archimedean, complete, singular Archimedean, and for orthocomplete l-groups. Andersen, Conrad, and Kenny [67] considered the connection between the splitting property and essential extensions of l-groups. A lattice ordered group H, GC.H, is called an essential extension of the l-group G is for each convex l-subgroup A, A ~ E, of the l-group H one has G,QA ~ E. An l-group which does not have proper essential extensions is called an essentially closed l-group. Earlier Conrad proved that any Archimedean l-group has a unique up to /-isomorphism es- sential closure. In [67] it is proved that an Archimedean l-group G has the splitting property if and only if G coincides with any of its essential Archimedean extensions in which it is an l-ideal. Any essentially closed Archimedean l-group has the splitting property.
We recall that by the polar X • of the set X of the l-group G is meant the set of all elements of G, orthog- onal to each element of X. Any polar of an l-group is a convex l-subgroup of it. A lattice ordered group G is called a P-group (SP-group), if each of its polars (the polar of any element g ~ e) is an l-direct factor of G. If H is an essential extension of the l-group G and H is a P-group (SP-group), where no proper l-subgroup of H containing G is a P-group (SP-group), then H is called the P-hull (SP-hull) of G. We recall that the l-group G is called O-approximable (or representable), if G is isomorphic with an t-subgroup of the Cartesian l-pro- duct of 1. o. groups. Bleier [79] proved that the 1-group G has a P (or SP)-hull if and only if G is O-approxim- able, while these hulls are unique. The SP-groups were also studied by Jakubik [137].
2629 Fleischer [106] considered decomposition into a Cartesian /-product of/-groups with operators. Miller [173] studied /-direct factors in /-groups. Jakubikova [147] considered/-groups G, in which each /-direct factor is isomorphic with G, and/-groups in which this property fails. Conrad [96] proved that a vector lat- tice G of countable dimension over an 1. o. field under certain assumptions can be approximated by order simple 1. o. spaces. In [98] there were considered essential extensions and automorphisms of vector lattices. Orthogonally Complete /-Groups. Various versions of the concept of orthogonatly complete were con- sidered in several papers. The most natural definition of an orthogonally complete l-group was given by Ber- nau first for P-groups, and later extended to arbitrary l-groups. Reducing the account to a unified termino- logy, we shall call an l-group orthocomplete if each set of positive pairwise orthogonal elements of G has in G a least upper bound. Bernau [74, 75] proved that for any l-group G there exist a unique up to l-isomorphism orthocomplete /-group H and l-isomorphism ~ :G --H, such that ~(G) is dense in H and between ~p(G) and H there are no orthocomplete I-subgroups. In this case H is called the orthocompletion of G. For O-approxim- able /-groups, Bleier [80] gave another construction of the orthocvmpletion. Bernau [74] proved that for Archimedean l-groups the operations of orthocompletion and Dedekind completion are interchangeable. Jaku- bik [132] established that if G is orthocomplete and any of its countable bounded subsets has a least upper bound in G, then G is complete, i.e., any bounded subset of G has .a least upper bound in G. An analogous result was obtained by Rotkovich [54]. In [55] it is proved that in Archimedean l-groups, from the existence in G of a least upper bound of any bounded set of pairwise orthogonal positive elements follows the orthocompleteness of G. For other connections between the concepts of completeness, conditional completeness, orthocompleteness, and Archimedeanness, cf. [53, 56, 141]. Complete l-Groups. Imbeddings. It is well known that any complete/-group is Archimedean. A sub- group H of the l-group G is called closed if for any subset M of H such that in G there exists a least upper hound SuPGM, there also exists SUPHM, while SuPGM = SUpHM. Jakubikova [148, 149] introduced the concept of a generating set in a complete l-group G, i.e., a subset X such that the closed convex l-subgroup of G generated by the set X coincides with G. She studied questions of existence and properties of free objects with two generators in the classes of complete, complete singular, complete without singular elements, complete orthocomplete, and complete completely distributive l-groups. In [113] under certain assumptions there was given a decomposition of a complete l-group into a product of convex l-subgroups of special form. Jakubik [132, 140] generalized the concept of Dedekind completion D(A), familiar for Archimedean l- groups A, to the case of an arbitrary/-group. He proved the existence in any/-group G of a largest Archi- medean convex l-subgroup A(G), which he calls the Archimedean kernel of G. By the generalized Dedekind completion Di(G) of the /-group G is meant an /-group such that: 1) G is an /-subgroup of Di(G); 2) D(A(G)) is an /-ideal in DI(G); 3) if x~G and X is a nonempty subset of xA(G), bounded in xA(G), then there exists x0~Di(G ) such that x0 = SupX; 4) for any k0~Dl(G) there exists xeG and XC_xA(G) such that x 0 = SupX. The exis- tence and uniqueness of DI(G) is proved for an arbitrary/-group G. If A(G) is closed in G, then D(A(G)) is closed in DI(G). Let ~ be any of the classes of/-groups: Archimedean, O-approximable, Abelian divisible. If G~, then DI(G)~. In [142] it is established that DI([IIGa) ~ iIlDl(Ga). In [141] it is proved that the classes of SP-groups and conditionally orthogonally complete l-groups are closed with respect to taking generalized Dedekind completions. The connection between higher degrees of distributivity for G and DI(G) is studied. In [139, 142] Jakubik introduced and studied the concept of maximal Dedekind completion and compared it with the concept of generalized Dedekind completion. Koldunov [27] considered the connection between Dedekind and O-completions of Archimedean /-groups. If in the l-group G the l-subgroup H is such that the map X -- Xf]H is a one-to-one mapping of the set of closed convex /-subgroups of G onto the set of closed (in H) convex /-subgroups of H, then G is called an ot*- extension of H. G is called ~*-closed if itdoes not have proper ~*-extensions. In [119] the existence was proved ofol *-closures for completely distributive, and in [70] for arbitrary /-groups. 5*-closures of l-groups were also studied in [82]. Bludov [5] constructed an example of an Archi- medean complete (i. e., not having proper linearly ordered essential extensions) 1. o. group, not all the factors of jumps of convex subgroups of which are isomorphic with the additive group of real numbers in the natural order. Essential extensions of/-groups were also studied by Kenny [151], Loonstra [159]. Conrad [94] consid- ered imbeddings of an Abelian l-group G in the group V(A) of all real-valued functions, defined on the root system A, and such that for any fEV(A) the set of those o eA such that f(o~) ;~ 0, satisfies the condition of maxi- mality. Other types of imbeddings, similar to those listed, were considered in [209, 90, 11, 187].
2630 By an amalgam (GI, G2, H, al, 0"2} is meant a collection of/-groups Gi, G2, H, and l-isomorphisms ~i:H~Gi (i = i, 2). One says that the class 3 of l-groups has the property of amalgamability, if for any amalgam 2631 In [81] it is proved that a free Abelian l-group is characteristically simple. A free l-group cannot con- tain an uncountable set of pairwise orthogonal elements. Free Abelian l-groups over distributive lattices were considered by Arkhipova [2, 3]. Torsion Classes. Radical Classes. ~,~artinez [162] called a torsion class a class ~ of l-groups, closed with respect to l-homomorphisms, taking /-subgroups, and such that in any /-group G there is contained a convex /-subgroup ~(G),belonging to R and containing all convex/-subgroups of G from the class ~. ~(G) is called the A-radical of G. The question of the possibility of reconstructing the torsion class from the ~ -radi- cal was solved positively by Jakubik [138]. The set of all torsion classes forms a complete Brauer lattice (Martinez [164]). A torsion class is called complete is it is closed with respect to extensions. Holland and Martinez [128] proved that complete and certain other torsion classes are join-irreducible. Holland [127] proved that any/-manifold is a torsion class. The following classes are torsion classes: hyperarchimedean l-groups [161, 95], the classes of all Cartesian l-products of 1. o. groups, Abelian 1. o. groups, 1. o. groups of integers. A large number of other torsion classes and relations among them are given in Conrad's survey [99] and in [134, 144, 165]. The class of Archimedean l-groups is not closed with respect to l-homomorphisms and hence is not a torsion class. Nevertheless, in any /-group G there exists an Archimedean kernel A(G), containing all Archi- medean convexl-subgroups of G. Properties of A{G) were studied in [83, 91, 92, 139, 140, 142, 152]. Algorithmic Questions in l-Groups. Although the class of all l-groups is a manifold of signature l, it turns out that the class of all groups, which admit a lattice order, nonaxiomatizable in the signature of the group (Vinogradov [14]). Holland and McCleary proved in [129] that the problem of equality of words in a free l-group is algorith- mically unsolvable. Glass [115] constructed an example of a recursively representable t-group with unsolvable word problem. Holland [126] considered the question of the equality to the identity of certain types of group words in a free product of l-groups. Jakubik [143] gave a new method for defining l-groups in terms of formal operations of signature l. Kutyev in [35, 36, 37] characterized /-groups in terms of the lattice of subsemi- groups. Isometrics in /-Groups. In [215, 216] there is introduced the concept of an isometry f in an Abelian /-group G as a mapping of G onto G such that for all x, yeG one has If(x)-f(y)l = Ix-yl. It was shown that for any isometry f of an Abelian l-group G one can find an isometric involutive automorphism T of the l-group G such that for any xeG one has f(x) = T(x) + f(o). In particular, the isometry f preserves order if and only if f is a translation of G: f(x) = x + f(o). By G* one denotes the group of all one-to-one isometrics of G with the order f_ 3. Groups of Automorphisms of Linearly Ordered Sets It is well known that the group Aut{X) of all automorphisms of the 1. o. set X is an l-group with respect to the following order: f _< g in Aut(X) if and only if f(x) -< g(x) for all x~X. Holland's theorem {1962) that any l-group has a faithful representation by automorphisms of a 1. o. set emerged as a powerful instrument for the study of l-groups and gave impetus to the intensive investigation of groups of automorphisms of 1. o. sets. The basic features of the theory of automorphism groups were developed at the end of the 1960s and the start of the 1970s, and cannot be treated completely in the present survey. Here we shall give only the main con- cluding results and their applications to certain questions of the theory of l-groups. Most of the concepts of the theory of representations of groups by permutations can be interpreted naturally in the group of automorphisms of a 1. o. set. We shall give some of these concepts which are needed in what follows. By the stabilizer StG (a) of the point a of the l.o. set X in the group of automorphisms GC__Aut(X) is meant the set of those g~G such that g(a) = a. By X(a, x) for a, x~X we denote the set {g(x)lg~StG(a)}. The 2632 group G of automorphisms of the 1. o. set X is called periodic if there exists an automorphism t of the Dede- kind closure X of the 1. o. set X, commuting elementwise with G on X such that t has no fixed points in X, and for any aEX one has {y~Xla < y In [168, 169] there is concluded a cycle of papers on transitive O-primitive l-groups of automorphisms of i.o. sets and the following classification theorem is proved. If G is a transitive O-primitive l-group of automorphisms of the 1. o. set X, then G is a group of one of the following types: 1) G is an Archimedean 1. o. group; 2) G is O-2-transitive on X, i.e., for any xl, x2, Yl, Y2 in X, xl < Yl, z2 < Y2, there exists a gEG such that g(xi) = x2, g(Yl) = Y2; 3) G is a periodic group of automorphisms of X, while the stabilizer of each point a of X in G acts faithfully on each set X(a, x) and is an O-2-transitive group of automorphisms of X(a, x), having an element with carrier bounded in X(a, x). In [72] there was studied the correspondence between right-regular representations of right-ordered groups and regular groups of automorphisms of 1. o. sets and the results obtained were applied to the descrip- tion of right-regular representations of right-ordered groups with subnormal systems of convex subgroups. Glass [114] gave new examples of simple l-groups, which are O-2-transitive groups of automorphisms without elements with bounded carriers, It is proved [167] that any l-automorphism of the l-group Aut(X) of the a-set X is inner. It is established [124] that if Aut(X) is transitive, then any l-automorphism of Aut(X) is induced by an inner automorphism of Aut(X). If Aut(X) is O,primitive, then any l-ideal of Aut(X) is charac- teristic. ~ely Distributive l-Groups of Automorphisms. In [168] it was established that an l-subgroup G of the l-group Aut(X) is completely distributive if and only if for any g, ga EG, a ~I, the equation g=Supcg~ implies g~SupAu~(x)g~. Let G be an l-subgroup of Aut(X), O(g, x) be the supporting interval of the element geG, xeX: O (g, x) = {yEXIgn(x) - y -< gin(x) for some integers n, m}. By a compression of O(g, x) is meant an element kg~Aut(X), such that kg(y) = y for y~O(g, x), kg(y) = g(y) for yeO(g, x). G is called a compressed /-subgroup of Aut(X) if for any geG, x~ kgeG. Read [198] noted that any compressed /-subgroup in Aut(X) is completely distributive and stabilizers of points in G are closed. In [119] there were studied a*-ext.ensions of completely distributive subgroups of Aut(X). Intransitive l-Groups of Automorphisms. McCleary in [170] carried over the basic concepts and theo- rems of the theory of transitive l-groups of automorphisms to the case of intransitive l-groups. He introduced the concept of convex O-block, studied the factorization of intransitive 1-groups in relation to the congruence defined by the partition of X into convex O-blocks. He studied O-primitive intransitive l-subgroups of Aut(X) and their connections with transitive O-primitive groups of automorphisms of 1. o. sets. The methods devel- oped are applied to the study of completely distributive l-groups and stabilizers of points. Groups of automorphisms of 1. o. sets were also studied in [123, 16, 49, 50, 51, 52]. 4. Topological /-Groups t/-Groups. If on the l-group G there is given a topology, with respect to which the group and lattice operations are continuous, then G is called an /-group. The topology of a tl-group is called convex if G has a basis of open sets consisting of convex subgroups of G. A closed (in the topological sense) subset of a tl- group will be called t-closed. Ball [71] proved that a convex l-subgroup of an O-approximable tl-group is closed if and only if it is t- closed with respect to any Hausdorff topology on G turning G into a tl-group. An O-approximablel-group has the weakest Hausdorff topology if and only if it is completely distributive. In [212] there were studied proper- ties of the lattice of all topologies on an l-group, turning it into a tl-group. Topological Properties of tl-Groups. One says that the tl-group G has bicompact origin if in G there is a neighborhood of the identity with bicompact t-closure, generating G. A. V. Mironov [41-43] proved that a nilpotent i.o. tl-group G of bicompact origin has a system G = H0 D HI D... D Hn = K D_ E of t-closed convex subgroups such that K is the connected component of the identity of G, which either coincides with E, or is 2633 tl-isomorphic with the additive group R of real numbers with the usual order and topology. G for any i = 0, 1 ..... n is a lexicoextension of Hi by the 1. o. group G/H i and each Hi/Hi+ 1 (0 <_ i _< n - 1) is isomorphic with a discrete finitely generated tl-subgroup of R. If G is not linearly ordered, then G has finite orthogonal rank and is obtained from a finite number of nilpotent 1. o. tl-groups of bicompact origin by successive appli- cati~)n of the direct product (in the topological and tl-group sense) and lexicographic extension. For Abelian /-groups the analogous result was obtained by A. N. Islamov [22, 23]. In [24] it is also established that if the Abelian tl-group G with convex topology and equal to UU~ for any neighborhood U of the identity has n=l finite orthogonal rank, then G is tl-isomorphic with all of R n. Khudaiberdiev [65] considered the connection between topologies and norms on l-groups. Pshenichnov [45-47] considered l-groups with a topology, contin- uous at the identity, where the order of'G is relatively closed. If G is locally bicompact, then its connected component is isomorphic with R n. A nilpotent l-group of bicompact origin with relatively closed order is a tl-group. The methods obtained are applied to the investigation of Lie groups with lattice order. 5. Ordered Groups Similar to /-Groups Riesz Groups. A partially ordered set G is called a TR(m, n)-set if it is directed and for any al, .... am, b 1..... bn~G such that for alli, j, a i 2634 Cyclic and Semihomogeneous Ordered Groups. The set G is called partially cyclically ordered if on G there is given a ternary relation R with the properties: 1) (a, b, c)~R implies a ~b ~c ~a; 2) (a, b, c)eR implies (a, c, b)~-R; 3) (a, b, c)~R implies (b, c, a)~R; 4) (a, b, c)s (a, c, d)~R implies (a, b, d)tR. If for anya, b, c~G, a ~b ~c ~a one of the relations (a, b, c)tR or (a, c, b)~R is true, thenG is called cycli- cally ordered. Zheleva [20] studied groups G with cyclic order R such that for any x, a, b, c~R the relation (a, b, c)~R implies (ax, bx, cx)~R. She calls such groups RCO-groups. Any right-orderedgroup turns naturally into an RCO-group. Any RCO-group is a quotient group of a suitable right-orderedgroup by some central cyclic subgroup. Each RCO-group can be represented by automorphisms of a suitable cyclically ordered set. Bludov and Kokorin [7] considered semihomogeneous lattice ordered groups, i.e., groups on which there is given a lattice order <__ such that for any giG, g ~ e, one of these properties is true: 1) a < b implies ag < bg, ga 6. Extensions of Partiai Orders Vinogradov [15] gave a sufficient condition for the existence on a three-stage solvable group of a non- trivial partial order. Mura and Rhemtulla [178] proved that in any torsion-free group from the manifold any maximal partial order is isolated. Extension of partial orders to interpolational orders was considered by Todorinov [59]. Mura and Rhemtulla [184] constructed examples of orderable nonhyperorderable groups with new interesting properties. Extension of partial orders in systems similar to groups was considered in [192] and [194]. 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In particular, in citing papers reflected in these surveys, the reader, as a rule, is referred to the corresponding survey paper. References to the bibliography of the surveys cited are for- mulated as [M62:000], [M65:000], [M68:000], [M71:000], [M75:000], and [KE:000], respectively. For exam- ple, [M65:121] means that one has in mind paper 121 from the survey M65 -- [131], relating to 1963-1965. References of the type Paragraph 3.1 indicate Section 3 Paragraph 1 of the present survey. In the bibliography as a rule, we do not include reports to conferences or seminars, if the corresponding results have already ap- peared in the form of journal publications. An exception is made only for survey reports. Definitions of con- cepts which are in the monographs of Lambek [M71:59] and Faith [161] are usually not given. Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 19, pp. 31-134, 1981. 2642 0090-4104/83/2306-264257o50 1983 Plenum Publishing Corporation