Rachůnek, Jiří

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJiří RachůnekZ-subgroups of ordered groupsMathematica Slovaca, Vol. 29 (1979), No. 1, 39--41Persistent URL: http://dml.cz/dmlcz/136197Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1979Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMath. Slovaca 29,1979, No. 1, 39—41 Z-SUBGROUPS OF ORDERED GROUPS JIRI RACHUNEK In this paper the concept of a z -subgroup of a lattice-ordered group is generalized for an ordered group (henceforth a po -group). Properties of z-subg-roups are investigated for the case of a 2-isolated regular po -group with the property (II). A po-group G = (G, + , = ) will be called 2-isolated if the holds: If aeG satisfies a = — a, then a = 0 . A po-group G is said to be regular if the existence of inf (a, b) in G+ implies the existence of inf (a, b) in G for a, b e G+. Way say that a po-group G has the property (II) if for each aeG there exists av—a = supG (a, —a). (See also [3].) We shall denote the set of all directed convex subgroups of a po-group G by r(G), the set of all convex subsemigroups with 0 of G+ by f(G). In [2, Theorems 2.1, 2.2, 2.3] it is proved that r(G), f(G) ordered by the inclusion form isomorphic complete lattices and that the infimum in f(G) is determined by the intersection. Let G be a po-group, au ..., an e G. We denote U(au ..., an) = {x e G; a, =x for all / = 1, ..., n), L(au ..., an) = {y eG; y=at for all / = 1, ..., n). For any element xeG we write |JC| = U(x, —x).x,yeG will be called disjoint (notation xdy) if there exist j ^ e |JC|, yxe \y\ such that JCi Ay! = 0. (jCiA^! means infG (xu yx).) For <£-£ A c G we denote A6 = {xeG; adx for all a e A } For JC e G we write x6 = {JC}6. If A6± <P, then A6 will be called a 5-polar of the set A . (See [3].) A66 means (A6)6. If A6± <P, then A c= A66, A6 = A666. For A6± 0>, B6±<P, A c B implies B6 cz A6. By [3, Proposition 2.5], any 6-polar of a 2-isolated po-group with the property (II) is an element of r(G). Moreover, the set of all <5-polars of a 2-isolated regular po-group with the property (II) ordered by the inclusion is a complete Boolean algebra and the infimum is formed by the intersection. ([3, Theorem 2.6].) Finally, if G is a po-group, then M e r(G) will be called a r-subgroup of G if for any a eM, b eG, a6 = b6 implies b eM. (For an /-group see e.g. [1].) 39 Theorem 1. Let G be a 2-isolated regular po-group with the property (II), MeT(G). Then the z-subgroup generated by M is M= V a66-a e M Proof. Let MeT(G), x, y eM. Then by [3, Lemma of Proposition 2.8] there exist elements a, beM+ such that xea66, yeb66, hence x-yea^vb33 = (a + b)33, therefore x—y eM, and so M is a subgroup of G. Let a eM( i . e . aeb33, where b eM), \x\^\a\. Then* v —x ^ av —a, and thus x3=>a3, hence x33^a33. Therefore xex33 cz a66 cz b33 holds, and this means xeM. Hence, by [3, Lemma 2 of Proposition 2.5], we obtain M e r ( G ) . Now, let x eM, y e G, x6 = y3. Then there exists aeM such that x e a66, thus y ey33 = x33 cz a33 holds, and consequently y eM. Therefore M is a z-subgroup of G. Let us show that M is the smallest z-subgroup of G containing M. Let us suppose that for a z-subgroup Z of G there holds MczZ . If O ^ a e M , O^xea 1 ^ (i.e. x eM+), then by [3, Proposition 2.8] (x + a)3 = x3na3 = a3, hence x + aeZ. Consequently O^x^x + a, x +a eZ, therefore by the convexity of Z we obtain xeZ. This implies M + c z Z + , thus also M c Z . Let G be a po-group. Then S ef(G) will be called a z-subsemigroup of G+ if xb =y* implies y eS for each x eS,y e G+. We denote the set of all z-subsemigr-oups of G+ by &(G), the set of all z-subgroups of G by %(G). In [2, Theorem 2.1] it is proved that the mapping cp\ F(G) —> f(G) given by Acp=A+ for each A eT(G) is an isomorphism between the sets F(G) and f(G) ordered by the inclusion and that Scp~l = (S) for each S e f(G), where (S) is the subgroup of G generated by S. Theorem 2. Let G be a 2-isolated regular po-group with the property (II). If Me%(G), then Mcpe&(G) and if S e£(G), then Sep'1 e%(G). Proof, a) Let MeZ£(G), xeM+, yeG+, x6=y\ Then yeM nG+ = M+, therefore Mcp e 2,(G). b) Let SeS(G), ue ( S ) , veG, u6 = vs. Then (u v - u f = (v v -v)6. Hereby uv —u e (S)+ = S, vv — v eG+, thus u v —v eS. And since — (v v — v) ^ v ^ v v — v, there holds (by the convexity of (S ) ) v e ( S ) . Therefore Sqp-1 e^(G). Theorem 3. If G is a 2-isolated regular po-group with the property (II), then 3£(G) andS(G) form isomorphic complete lattices that are closed A-subsemilat-tices of r(G) and f(G), respectively. Proof. Let S, e&(G) (iel), S=n S,. If x e ( S ) , then x63 cz (S,) for each iel, iel and since x33 and (S.) belong to F(G), ( 0 + c z S , f o r e a c h / e F Hence (x<v>)+czS. But this means that x33 cz ( S ) , thus (S) e&(G). Therefore S e£(G). And since G+ e££(G), then S(G) is a complete lattice. The rest is evident. 40 In [3, Corollary 1 of Proposition 1.2] it is proved that for each positive element a of a 2-isolated po -group G, the smallest directed convex subgroup of G containing a is C(a) = {x e G; \x\ ^ \na\ for a positive integer n}. If G has also the property (II) and if b is an arbitrary element of G, then each directed convex subgroup B of G containing b contains bv —b, too, therefore B 3 C(b v — b). It follows that the smallest directed convex subgroup C(b) containing b is equal to C(bv —b). Theorem 4. Let Gbea 2-isolated regular po-group with the property (II). Then the following are equivalent: (1) r(G) = 2£(G). (2) C(a) = a66 for each a e G. (3) C(a) = C(b) if and only if a66 = b66 for each a,beG. Proof. 1 => 2 : Let a e G. By the assumption C(a)e2t(G), hence a66 c C ( a ) . And since a66 eT(G), C(a)^a66always holds. 2 - ^ 3 : Trivial. 3 => 1: Let MeT(G), aeM, beG, a6 = b6. Then a66 = 66 6 , thus beC(b) = C(a)c=M. REFERENCES [1] BIGARD, A.: Contribution à la théorie des groupes гeticules, These Fac. Sci. Paгis, 1969. [2] RACHÜNEK, J.: Directed convex subgгoups of ordered groups, Acta Univ. Palack. Olomucensis, Fac. Reг. Nat., 41, 1973, 39—46. [3] RACHÜNEK, J.: Prime subgroups of oгdered groups, Czechoslov. Math. J., 24 (99), 1974, 541—551. Received March 10, 1977 Kаtedга аlgebгy а geometríe Příгodovědecké fаkulty Univeгzity Pаlаckého Leninovа 26 771 46 OІomouc 2.-ПОДГРУППЫ УПОРЯДОЧЕННЫХ ГРУПП Йиржи Рахунек Резюме В статье обобщено понятие г -подгруппы из теории решёточно упорядоченных групп для любых упорядоченных групп. В частности, здесь показаны некоторые основные свойства I -подгрупп в случае 2-изолированных регулярных упорядоченных групп, в которых существует вир (ау —а) для длюбого элемента а. 41 

$Z$-subgroups of ordered groups

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$Z$ -subgroups of ordered groups

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ZZZ-subgroups of ordered groups

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$Z$ -subgroups of ordered groups