Kepka, T.

Niemenmaa, M.

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

Acta Universitatis Carolinae. Mathematica et PhysicaTomáš Kepka; Markku NiemenmaaOn splitting k-systemsActa Universitatis Carolinae. Mathematica et Physica, Vol. 27 (1986), No. 2, 3--10Persistent URL: http://dml.cz/dmlcz/142568Terms of use:© Univerzita Karlova v Praze, 1986Institute 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 mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.cz1986 ACTA UNIVERSITATIS CAROLINAE—MATHEMATICA ET PHYSICA VOL. 27. NO. 2 On Splitting k-Systems T. KEPKA Department of Mathematics, Charles University, Prague M. NIEMENMAA Department of Mathematics, University of Oulu Received 15 November, 1985 A connection between groups and certain groupoids is investigated. Vysetfuje se souvislost mezi grupami a jistymi grupoidy. H3ynaeTCH CBH3B rpyrm H HeKOTOpbix rpyimoHflOB. 1. Introduction Generalizations of the notion of the arithmetic mean value have been considered e.g. in [2] and [5]. If (G, + , •) is an algebra where (G, + ) is a uniquely 2 — divisible abelian group and the multiplication is the usual arithmetic mean value, then x + + yz = xy + xz. The aim of the present paper is to study algebraic systems satis-fying identities of a similar type. Let k be an integer. An algebra (G, + , •) with two binary operations is called a k-system if (G, + ) is a group (possibly non-commutative) and the following ident-ity holds: (1) x + k(yx) = xy + xz . Let G = (G, + , •) be a k-system. We denote by Z{G) the centre of the group (G, +) . If a e G then we have four transformations Lfl, Ra, Lfl and Ra of G defined by La(b) = ab, Rjb) = ba, L+(b) = a + b and Ra(b) = b + a. Obviously, both La and Ra are permutations. Further, we define five transformations d, e, f, g and h of G by d(a) = aa, e(a) = (2 — k) a, f(a) = ka, g(a) = (k — 1) a and h(a) = 2a for every a e G. In the sequel we shall need the following simple result. 1.1. Lemma. Let k + 0 be an integer and (G, + ) a 2-divisible group such tha* ka = 0 for every a e G. Then (G, + ) is uniquely 2-divisible. *) 186 00 Praha, Sokolovská 83, Czechoslovakia **) 90570 Oulu, Finland 1.2. Example. Let (G, + ) be a 2-divisible group and let t be any transformation of G such that 2 t(a) = a for every a e G. Put ab = f(a) for all a,beG. Then (G, + , •) is a 0-system. 1.3. Example Let (G, + ) be a 2-divisible abelian group and let t be any trans-formation of G such that 2 f'a) = a for every a e G. Put ab = t[a) + f(b) for all a, b e G. Then (G, + , •) is a 1-system. 1.4. Example. Let fc be an integer and (G, +) a group such that fca = 0 for every ae G. Suppose further that the group (G, + ) is uniquely 2-divisible (e.g. fc is odd or fc =j= 0 and (G, + ) 2-divisible — see 1.1) and put ab = a/2 for all a, be G. Then (G, + , •) is a fc-system. The fc-systems constructed in 1.4 will be called fc-systems of type one. 1.5. Example. Let fc be an integer and (G, + ) an abelian group such that (fc — 1) a = 0 for every a e G. Suppose further that the group (G, + ) is uniquely 2-divisible (e.g. fc even or fc 4= 1 and (G, + ) 2-divisible — see 1.1) and put ab = = a/2 + 6/2 for all a,beG. Then (G, + , •) is a fc-system. The fc-systems constructed in 1.5 will be called fc-systems of type two. 1.6. Example. Let fc be an integer and (G, + ) a group such that fc(fc — 1) a = 0 and fca eZ(G) for every a e G. Suppose further that the group (G, + ) is uniquely 2-divisible (e.g. fc =f= 0, 1 and (G, + ) 2-divisible - see 1.1) and put ab = a/2 + fcb/2 for all a,b eG. Then (G, + , •) is a fc-system. A fc-system G is said to be splitting if it is isomorphic to the direct sum of a fc-system of type one and a fc-system of type two. Finally, the reader should consult [1] and [4] for definions, notation etc. 2. Basic properties of k-systems In this section let G be a fc-system. 2.1. Lemma. For all a,beG: (2) a = (2 - fc) (aa), (3) (fc - 1) (ab) = - a + + aa = (fc - 1) (aa), (4) 2(ab) + (2 - 2k) (bb) = a + b, (5) 2(ab) = a + fc(bb). These equalities are easy consequences of (1). 2.2. Lemma. fc(fc — 1) a = 0 for every a e G. Proof. We have fc(fc - 1) (aa) = (fc - 1) (0 + fc) (aa)) = (fc - 1) 2(0a) = = 2(fc - 1) (0a) = 2(00) by (3) and (5). Hence, by (2), fc'vfc - 1) a = k[k - 1) . . (2 - fc) (aa) = 2(2 - fc) (00) = 0. 2.3. Lemma, (i) ed = idG, (ii) gLa is constant for every a e G, (iii) Ra = = R-hgd(a)hRa a n d RHaa) = hRa for every a e G, (iv) L+JLb = LabLa for all a,beG, (v) L+afRb = R:bLa for all a, b e G. Proof, (i) follows from (2), (ii) from (3), (iii) from (4) and finally (iv) and (v) follow from (1). 2.4. Lemma. The transformation e is surjective, d is injective, h is surjective and Ra is injective for every as G. Proof. Use 2.3. 2.5. Corollary, (i) The group (G, + ) is 2-divisible and (fe - 2)-divisible, (ii) The group (G, +) is uniquely 2-divisible provided that fe 4= 0,1, (iii) Let fe 4= 0, 1 and fe(fe — 1) = 2% i ^ 0, j odd. Then ja = 0 for every a e G. (iv) The groupoid (G, •) is right cancellative. 2.6. Proposition. The following conditions are equivalent: (i) The groupoid (G, •) is right divisible. (ii) The groupoid (G, •) is a right quasigroup. (iii) The group (G, +) is uniquely 2-divisible. Furthermore, these conditions are satisfied if fe + 0, 1, Proof. See 2.3 (iii) and 2.5. Now fe-systems satisfying the equivalent conditions of 2.6 are said to be regular. 2.7. Lemma. Let a, b9 c e G be such that ab = ac. Then fLb = fLc9 fRb = fRc and f(bb) = fKcc). Proof. By 2.3 (iv), L+afLb = L+abLa = L+acLJLC9 so that fLb=fLc. Similarly fRb = fRc. Combining the two equations we get f(bb) = fyce). 2.8. Lemma. For all a9 b e G: (i) k(aa) = ka and (ii) 2(ab) = a + kb. Proof, (i) We have ka = k{2 - fe) (aa) = (2 - fe) k(aa) = 2k(aa) - k\aa). By 2.2, k2(aa) = k(aa) and hence ka = k(aa). (ii) This follows from (i) and (5). 2.9. Lemma. 2(00) = 0 and if fe = 0, then 00 = 0. Proof. By 2.8 (ii), 2(00) = 0. If fe + 0, 1 then (G, +) is uniquely 2-divisible and 00 = 0. If fe = 1, then 00 = 0 by (2). 2.10. Lemma. Let G be regular. Then fee and fec/2 are elements of Z(G) for every ceG. Proof. Put q = h~x. By 2.8 (ii), ab = q(a + kb) for all a, b e G. Let ceG. Now the equation (l) can be written in the form: a + kq(b + fee) = q(a + kb) + q(a + fee) . For a = 0 we have kq(b + fee) = q(kb) + q(kc) , so that a + q(kb) + q(kc) = q(a + kb) + q(a + fee) . From this, for c = 0, we see that a + q(kb) = q(a + kb) + q(a) , hence a + q(kb) — q(a) = q(a + kb) , and therefore a + q(kb) + q(kc) = a + q(kb) — q(a) + a + q(ke) — q(a) . Consequently, q(kc) = q(a) + q(kc) - q(a) , so that q(kc) e Z(G). 3. 0-systems 3.1. Proposition. Let G be a 0-system. Then there exists a transformation t of G such that 2 t(a) = a and afe = t(a) for all a, b e G. Proof. It is enough to put t(a) = aa for every ae G. Now the result follows from (2) and (3). 4. 1-systems In this section let G be a 1-system. 4.1. Lemma, (i) a = aa and 2(ab) = a + b for all a, b e G. (ii) / = idG. (iii) L+Lb = L+abLa for all a, b e G. (iv) L ^ = R+hLa for all a, b e G. (v) L0 = R0. (vi) L^L, = Ka+0Lfl for every a e G. Proof, (i) follows from (2) and (5), (ii) is obvious, (iii) follows from 2.3 (iv) and (iv) from 2.3 (v). To prove (v) notice that we have L0 = R0L0 = R00L0 = = L0R0 = R0. Finally (vi) is clear from (iii), (iv) and (v). 4.2. Lemma. Let a e G such that 0a e Z(G). Then La = Ra. Proof. By 4.1 (iii) and (iv), La = L+0aL0 = R+aL0 = Ra. 4.3. Lemma, a 0 6 Z(G) for every a e G. Proof. By 4.1 (vi), a 0 + a b = ab + a0 for each b e G. Hence, by 4.1 (i), aO + a + b = aO + 2(ab) = 2(ab) + aO=b + aQ. 4.4. Lemma. Both the group (G, + ) and the groupoid (G, •) are commutative. Proof. By 4.1 (v) and 4.3, a 0 = 0a e Z(G) for every a e G. Now (G, •) is com-mutative by 4.2 and (G, + ) by 4.1 (i). 4.5. Proposition. (G, + ) is an abelian group and there exists a transformation t of G such that 2 t(a) = a, t(0) and ab = t(a) + t(b) for all a,beG. Proof. Put t(a) = a 0. Then 2 t(a) = a by 4A (i), t(0) = 0 by 4A (i) and t(a) + + t(b) = Oa + Ob = 0 + ab = ab. 5. k-systems of type 1 5.1. Proposition. The following conditions are equivalent for a fc-system G: (i) ka = 0 for every a e G. (ii) ab = ac for all a,b,ce G. (iii) Either fc = 0 or G is of type 1. Proof, (i) implies (ii): This follows immediately from (3). (ii) implies (i): This follows easily from 2.8 (ii). (i) implies (iii): Suppose that fc 4= 0. Then (G, + ) is uniquely 2-divisible and ab = a/2 by 2.8 (ii). Thus G is of type one. (iii) implies (i): This is trivial. 5.2. Proposition. Now a 0-system is of type one iff it is regular. Proof. Use 3.L 5.3. Lemma. Let fc = ±2\ i _ 0. Then every fc-system of type one is trivial (i.e. a one-element set). Proof. Obvious. 6. k-systems of type 2 6.1. Proposition. The following conditions are equivalent for a fc-system G: (i) (fc — 1) a = 0 for every a e G. (ii) The groupoid (G, •) is idempotent. (iii) The groupoid (G, •) is commutative, (iv) Either fc = 1 or G is of type 2. Proof, (i) is equivalent to (ii): This is clear from (2). (i) implies (iv): suppose that fc #- 1. Then G is regular and (G, + ) is abelian by 2A0. Now ab = a/2 + 6/2 by 2.8 (ii), so that G is of type 2. (iv) implies (iii): This is obvious, (iii) implies (i): By 2.8 (ii), a + kb = 2(ab) = 2(ba) = b + ka for all a,beG. Thus for b = 0 we have (fc — 1) a = 0. 6.2. Proposition. Now a 1-system is of type 2 iff it is regular. Proof. Use 4.5. 6.3. Lemma. Let fc = +21, f _• 0. Then every fc-system of type 2 is trivial. Proof. Obvious. 7. Several consequences 7.1. Proposition. Let G be a k-system. Then ka e Z[G) for every a e G. Moreover, if G is regular then ka/2 e Z(G). Proof. The assertion is trivial for k = 0 and it follows from 4.5 for k = 1. If k 4= 0, 1 then G is regular and 2.10 can be applied. 7.2. Theorem. Let G be a regular k-system. Then ab = a/2 + kb/2 for all a, beG. Proof. The statement follows easily from 2.8 (ii) and 7.1. 7.3. Proposition. Let G be a k-system and H = {a e G: (k — 1) a = 0}. Then: (i) (H, + ) is a normal subgroup of (G, +) , H i= Z(G) and ka e H for every a e G. (ii) H is a subsystem of the k-system G and H is a k-system of type 2 provided that either k =j= 1 or G is regular. Proof. If k = 0, 1 then the situation is clear. Now assume that k + 0, 1. By 7.1 and 2.2, H c Z(G) and ka e H for every ae G. Thus (H, + ) is a normal subgroup of (G, +) . By 1.1, the group (G, +) / (# , + ) is uniquely 2-divisible and we see that a/2 e H for every a e H. The rest is clear from 7.2. 7.4. Proposition. Let k + 0 and let G be a k-system. Then: (i) Z G) is a subsystem of G. (ii) Z(G) is an abelian k-system. Proof. We can proceed similarly as in the proof of 7.3. 7.5. Proposition. Let G be a k-system. Define a relation r on G by (a, b) e r iff a — b e H. Then (i) r is a congruence of the k-system G. (ii) H is one of the blocks of G. (iii) The factor system G\r is a k-system of type one provided k 4= 0. Proof. We can assume that k + 0, 1. Now let a, b, ce G with (b, c)e r. Then b - ceH and ab - ac = a/2 + kb/2 - kc/2 - a/2 = kb/2 - kc/2. Further, (k - 1) (kb/2 - kc/2) = 0. We see that ab - ac e H, so that (ab, ac) e r. Similarly, ba - ca = 6/2 - c/2 = (b - c)/2 + c/2 - c/2 = (b - c)/2. Thus ba - caeH and (ba, ca) e r. We conclude that r is a congruence of the k-system G. The rest is clear. 7.6. Proposition. Let G be a regular k-system. Then thegroupoid (G, •) is medial (i.e. satisfies the identity xy . uv = xu . yv). Proof. Apply 7.1 and 7.2. 8. Some k-systems 8.1. Proposition, (i) Every 2-system is trivial, (ii) Let i _ 1. Then every +2f-system is of type 2. (iii) Le: i = 1 and k = ±2l + 1. Then every k-system is of type 1. Proof (i): This is clear from (2). (ii): Let G be a + 2£-system. Consider the con-gruence r of G by 7.5. From 5.3 and 7.5 (ii) it follows that r = G x G, i.e. H = G and G is of type 2. (iii): Using 6.3, we can proceed in the same way as in the proof of(ii). 9. Splitting k-systems In this section, let fe + 0, 1 and let G be a fe-system. Put K = {a e G: ka = 0} and define a relation s on G by (a, b) e s iff a — b e K. 9.1. Lemma, (i) (fe — 1) a e K for every ae G. (ii) HnK = 0,H. + K = G and each element g of G has a unique expression of the form g = h + x, where he H and x e K. (iii) 5 is reflexive and symmetric. (i3) r n s = idG. Proof. Now (i), (iii) and (iv) are clearly true. As for (ii), it is easy to see that H n K = 0 and H + K = G. Let g = h^ + xx = h2 + x4, where ht e H and xteK (i = 1, 2). Then hx - h4 = x2 - x t e H c Z(G). Thus 2(x4 - xx) = = (x2 — xx) + (x2 — xx) = x2 + (x2 — xx) - xx = 2x2 — 2xx. By induction, fe(x2 — xx) = fex4 W kxx = 0, hence x2 — x±e K. Now x2 — xx e H n K = 0, so that x2 = x t and h^ = h2 yeilding the uniqueness of the expression. 9.2. Theorem. If G is a fc-system, then it is splitting. Proof. Define a mapping m: G -> H © GJH (direct sum) by m(g) = (h, x + H), where g has the unique expression g = h + x[h e H, xe K). This mapping is a group homomorphism with Ker (m) = 0 and Im (m) = H © G/H. Hence (G, + ) ^ ( H , + ) © ( G , +) / (# , + ) and now it is clear that G is splitting by 7.3 and 7.5. 9.3. Corollary. If G is a fe-system, then (i) (K, + ) is a normal subgroup of (G, +) . (ii) The transformation / is an endomorphism of (G, +) . (iii) 5 is a congruence of the fe-system G. Proof. Now (i) and (ii) follow from 9.2 and (iii) is a consequence of 7.2. 10. Finite k-systems 10.1. Proposition. Let G be a finite fe-system of order n. Then n is odd and G is splitting. Proof. Use 2.6 and 9.3 (the cases fe = 0, 1 are trivial). Now let G be a finite 1-system of odd order n. By 4.4 it is easy to see that (G, •) is a commutative quasigroup. Thus the transformations La = Ra(a e G) are permuta-tions of G and they generate the multiplication group of (G, •) which we denote by M(G, •). Denote by t the mapping x -> ((n + l)/2) x. Clearly, t is an automorphism of (G, +) . Now we prove 10.2. Proposition. Let G be a finite 1-system with odd order n = p\x ... pnn (the primary decomposition of n, n odd.). Then M(G, •) is the group theoretical splitting extension of (G, •) by <t>. Furthermore, o(t) | 1 cm . (p? - par\ ..., Pann - 1C"1) • Proof. The first part of the proposition follows from lemma 2.2 in [3]. Since (n + l)/2 is a unit in zn and n is odd, the well-known properties of the group of units of zn imply the second past. References [1] BRUCK, R. H.: A survey of binary systems (Springer, Berlin 1971). [2] JEZEK, J., KEPKA , T., NEMEC, P.: Distributive groupoids. Rozpravy Ceskoslovenske Akademie Ved91/3 (Piaha, 1981). [3] KEPKA, T.: Multiplication groups of some quasigroups. Universal Algebra (Proc. Conf. Esztergom, 1977). Colloq. Math. Soc. Janos Bolyai 29 (North-Holland, Amsterdam 1982), 4 5 9 - 4 6 5 . [4] ROSE, J.: A course in group theory (Cambridge University Press, Cambridge 1978). [5] SOUBLIN, J. P.: Etude algebrique de la notion de moyenne. J. Math. Pures et Appl. 50 (1971), 5 3 - 2 6 4 . 10 

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On splitting $k$-systems

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On splitting $k$ -systems

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On splitting kkk-systems

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On splitting $k$ -systems