We give new equations that axiomatize the variety of trimedial quasigroups.
We also improve a standard characterization by showing that right semimedial,
left F-quasigroups are trimedial.Comment: 6 pages, AMS-LaTeX. To appear in Comment. Math. Univ. Carolinae. for
  a special issue: the Proceedings of Loops03. Version 3: the proof of the main
  result is collected together more formally; other stylistic change

Kinyon, Michael K.

Phillips, J. D.

English

arXiv

summary:We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial

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Axioms for trimedial quasigroups

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Commentationes Mathematicae Universitatis CarolinaeMichael K. Kinyon; Jon D. PhillipsAxioms for trimedial quasigroupsCommentationes Mathematicae Universitatis Carolinae, Vol. 45 (2004), No. 2, 287--294Persistent URL: http://dml.cz/dmlcz/119458Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2004Institute 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.czComment.Math.Univ.Carolin. 45,2 (2004)287–294 287Axioms for trimedial quasigroupsMichael K. Kinyon, J.D. PhillipsAbstract. We give new equations that axiomatize the variety of trimedial quasigroups.We also improve a standard characterization by showing that right semimedial, leftF-quasigroups are trimedial.Keywords: trimedial quasigroup, F-quasigroup, semimedial quasigroupClassification: 20N051. IntroductionA quasigroup Q = (Q; ·, \, /) is a set Q with three binary operations ·, \, / :Q × Q → Q satisfying the equations:x\(x · y) = y (x · y)/y = xx · (x\y) = y (x/y) · y = x.Basic references for quasigroup theory are [1], [5], [6], [14].A quasigroup is medial if it satisfies the identity(M) xy · uv = xu · yv.A quasigroup is trimedial if every subquasigroup generated by three elements ismedial. Medial quasigroups have also been called abelian, entropic, and othernames, while trimedial quasigroups have also been called triabelian, terentropic,etc. (See Chapter IV of [6], especially p. 120, for further background.) Theclassic Toyoda-Bruck theorem asserts that every medial quasigroup is isotopic toan abelian group [15], [4]. This result was generalized by Kepka to trimedialquasigroups: every trimedial quasigroup is isotopic to a commutative Moufangloop [7].There are two distinct, but related, generalizations of trimedial quasigroups.The variety of semimedial quasigroups (also known as weakly abelian, weaklymedial, etc.) is defined by the equationsxx · yz = xy · xz(Sl)zy · xx = zx · yx.(Sr)288 M.K.Kinyon, J.D.PhillipsA quasigroup satisfying (Sl) (resp. (Sr)) is said to be left (resp. right) semimedial .Every semimedial quasigroup is isotopic to a commutative Moufang loop [7]. (Inthe trimedial case, the isotopy has a more restrictive form; see the cited referencesfor details.)The variety of F-quasigroups was introduced by Murdoch in [13], the samepaper in which he introduced what we now call medial quasigroups. F-quasigroupsare defined by the equationsx · yz = xy · (x\x)z(Fl)zy · x = z(x/x) · yx.(Fr)A quasigroup satisfying (Fl) (resp. (Fr)) is said to be a left (resp. right) F-quasigroup. Murdoch did not actually name the variety of F-quasigroups. Wethank one of the referees for suggesting that the earliest use of the name mightbe in [2].One among many links between these two generalizations of trimedial quasi-groups is the following ([9, Proposition 6.2]).Proposition 1.1. A quasigroup is trimedial if and only if it is a semimedial, left(or right) F-quasigroup.Together with Kepka, we have been investigating the structure of F-quasi-groups, and have shown that every loop isotopic to an F-quasigroup is Moufang.Full details will appear elsewhere [10]. As part of that investigation, we were ledto consider the following equations, which are similar in form to (Fl), (Fr):x · yz = (x/x)y · xz(El)zy · x = zx · y(x\x).(Er)The main result of the present paper is the following.Theorem 1.2. A quasigroup is trimedial if and only if it satisfies (El) and (Er).Kepka [7], [8] showed that the variety of trimedial quasigroups is axiomatizedby the semimedial laws (Sl), (Sr), and by the equation (x ·xx) ·uv = xu · (xx · v).Later [11] we showed that (Sl) is redundant. Theorem 1.2 offers a more symmetricalternative.As an auxiliary result, we will also use (El) and (Er) to obtain the followingimprovement of Proposition 1.1.Theorem 1.3. Let Q be a quasigroup. The following are equivalent:1. Q is trimedial;2. Q is a right semimedial, left F-quasigroup;3. Q is a left semimedial, right F-quasigroup.Our investigations were aided by the equational reasoning tool OTTER devel-oped by McCune [12]. We thank T. Kepka for suggesting that (El), (Er) mightaxiomatize an interesting variety of quasigroups; he was certainly correct.Axioms for trimedial quasigroups 2892. ProofsOur strategy for proving Theorem 1.2 is to use Proposition 1.1: we will showthat a quasigroup satisfying (El), (Er) is a semimedial, F-quasigroup. First weintroduce some notation for local right and left unit elements in a quasigroup:e(x) := x\x f(x) := x/x.If Q = (Q; ·, \, /) is a quasigroup, then so are the left parastrophe Ql :=(Q; \, ·, /op), the right parastrophe Qr := (Q; /, ·, \op), and the opposite para-strophe Qop := (Q; ·op, /, \), where for a binary operation ∗, we use ∗op to de-note the opposite operation. Note that (Ql)l = Q, (Qr)r = Q, (Qop)op = Q,(Qop)l = (Qr)op, and (Qop)r = (Ql)op. For a more complete discussion of para-strophes, including alternative notation conventions, see [1], [14].We state the following obvious result formally for later ease of reference.Lemma 2.1. Let Q = (Q; ·, \, /) be a quasigroup.1. Q satisfies (Fl) if and only if Qop satisfies (Fr).2. Q satisfies (Sl) if and only if Qop satisfies (Sr).3. Q satisfies (El) if and only if Qop satisfies (Er).Parts (1) and (2) of the following lemma are well-known, although the authorshave not been able to find a specific reference.Lemma 2.2. Let Q = (Q; ·, \, /) be a quasigroup.1. Q is a left F-quasigroup if and only if Ql is left semimedial.2. Q is a right F-quasigroup if and only if Qr is right semimedial.3. Q satisfies (El) if and only if Ql satisfies (El).4. Q satisfies (Er) if and only if Qr satisfies (Er).Proof: For (1): In Ql, (Sl) is e(x)\(y\z) = (x\y)\(x\z). Multiply on the left bye(x), replace y with xy, and z with xz to get (xy)\(xz) = e(x)(y\z). Now replacez with yz and multiply on the left by xy to get x · yz = xy · e(x)z, which is (Fl)in Q. Since (Ql)l = Q, the converse also holds.For (2): By part (1), (Fl) holds in Qop iff (Sl) holds in (Qop)l = (Ql)op. Thedesired result now follows from Lemma 2.1.For (3): In Ql, (El) is ((x/opx)\y)\(x\z) = x\(y\z). Multiply on the left byf(x)\y, and replace z with yz to get x\(yz) = (f(x)\y)(x\z). Multiply on theleft by x, replace y with f(x)y, and z with xz to get f(x)y · xz = x · yz, which is(El) in Q. Since (Ql)l = Q, the converse also holds.For (4): By part (3), (El) holds in Qop iff (El) holds in (Qop)l = (Qr)op. Thedesired result now follows from Lemma 2.1. 290 M.K.Kinyon, J.D.PhillipsLemma 2.3. Let Q be a quasigroup.1. If Q satisfies (El), then f : Q → Q is an endomorphism of Q.2. If Q satisfies (Er), then e : Q → Q is an endomorphism of Q.3. If Q is a left F-quasigroup, then e : Q → Q is an endomorphism of Q.4. If Q is a right F-quasigroup, then f : Q → Q is an endomorphism of Q.Proof: In each case, it is enough to show that the multiplication is preserved.For (1): f(x)f(y) · xy = f(x)x · f(y)y = xy, and so f(x)f(y) = (xy)/(xy) =f(xy).For (2): Since f : Q → Q is an endomorphism of Q iff e : Q → Q is anendomorphism of Qop, this follows from part (1) and Lemma 2.1(3).For (3): if (Fl) holds, then xy · e(x)e(y) = x · ye(y) = xy, and so e(x)e(y) =(xy)\(xy) = e(xy) ([2, p. 38, equation (32)], [9, Lemma 4.2], [3]).For (4): This follows from part (3) and Lemma 2.1(1). Lemma 2.4. Let Q be a quasigroup. If e : Q → Q or f : Q → Q is anendomorphism of Q, then f(e(x)) = e(f(x)) for all x ∈ Q.Proof: If f is an endomorphism, then f(e(x)) = f(x)\f(x) = e(f(x)), and thecase where e is an endomorphism is similar. Lemma 2.5. Let Q be a quasigroup.1. If Q satisfies (El), then Q is a left F-quasigroup if and only if it is leftsemimedial.2. If Q satisfies (Er), then Q is a right F-quasigroup if and only if it is rightsemimedial.Proof: For (1): Assume Q satisfies (El). By Lemma 2.2(3), (El) holds in Ql.We will prove the implication (Sl) =⇒ (Fl). Since this will also hold in Ql, itwill follow from Lemma 2.2(1) that the implication (Fl) =⇒ (Sl) will hold in Q.Now if Q is left semimedial, thenxx · yz = xy · xz by (Sl)= f(xy)x · (xy · z) by (El)= (f(x)f(y) · x) · (xy · z) by Lemma 2.3= (x · f(y)e(x)) · (xy · z) by (El)= xx · (f(y)e(x) · (x\(xy · z))) by (Sl).Cancelling and replacing z with e(x)z, we havef(y)e(x) · (x\(xy · e(x)z)) = y · e(x)z= f(y)e(x) · yz by (El).Axioms for trimedial quasigroups 291Cancelling, we obtain yz = x\(xy · e(x)z) or x · yz = xy · e(x)z, which is (Fl).For (2): If Q satisfies (Er), then Qop satisfies (El) by Lemma 2.1(3). By part(1), (Fl) and (Sl) are equivalent in Qop, and so (Fr) and (Sr) are equivalent inQ by Lemma 2.1(1), (2). Lemma 2.6. A quasigroup satisfying (El), (Er) is an F-quasigroup.Proof: We will show (El), (Er) =⇒ (Fl). Since this implication will also hold inthe opposite parastrophe (by Lemma 2.1(3)), (Fr) will follow from Lemma 2.1(1).Thus we computex · yz = f(x)y · xz by (El)= [f(x)e(f(x)) · x(x\y)] · xz= [x · e(f(x))(x\y)] · xz by (El)= [x · xz] · [e(f(x))(x\y) · e(xz)] by (Er)= [x · xz] · [f(e(x))(x\y) · e(x)e(z)] by Lemmas 2.3 and 2.4= [x · xz] · [e(x) · (x\y)e(z)] by (El).Now xz · (x\y)e(z) = yz by (Er), and sox · yz = [x · xz] · e(x)[(xz)\(yz)].Now replace y with (xz · y)/z to obtainx(xz · y) = [x · xz] · e(x)y.Finally replace z with x\z to getx · zy = xz · e(x)y,which is (Fl). We can now prove our main result.Proof of Theorem 1.2: Suppose Q is trimedial. Since each of (El) and (Er)is a special case in three variables of the medial identity (M), these identities willhold in Q. Indeed, fix a, b, c ∈ Q. Then the subquasigroup 〈a, b, c〉 generated by{a, b, c} is medial. Taking x = f(a), y = a, u = b, v = c in (M), we obtaina · bc = f(a)b · ac, while taking x = c, y = b, u = a, v = e(a) in (M) givescb · a = ca · (b · e(a)). Since a, b, c ∈ Q were chosen arbitrarily, we have (El)and (Er).Conversely, if Q satisfies (El) and (Er), then by Lemma 2.6, Q is an F-quasigroup, and by Lemma 2.5, Q is semimedial. Proposition 1.1 completes theproof. 292 M.K.Kinyon, J.D.PhillipsLemma 2.7. 1. A right semimedial, left F-quasigroup satisfies (Er).2. A left semimedial, right F-quasigroup satisfies (El).Proof: For (1): Suppose Q satisfies (Sr) and (Fl). Then(xz · y) · e(xz)z = xz · yz by (Fl)= xy · zz by (Sr)= ((xy)/e(xz))e(xz) · zz= ((xy)/e(xz))z · e(xz)z by (Sr).Cancelling and using Lemma 2.3(3), we have xz · y = ((xy)/e(x)e(z))z. Nowx(y/e(z)) · e(x)e(z) = xy by (Fl), and so xz · y = x(y/e(z)) · z. Replacing y withye(z), we obtain xz · ye(z) = xy · z, which is (Er).For (2): If (Sl) and (Fr) hold in Q, then (Sr) and (Fl) hold in Qop(Lemma 2.1(1), (2)), and so (Er) holds in Qop by part (1). By Lemma 2.1(3),(El) holds in Q. We now turn to our auxiliary result.Proof of Theorem 1.3: Let Q be a right semimedial, left F-quasigroup. ByLemma 2.7(1), (Er) holds. By Lemma 2.5(2), Q is an F-quasigroup. We will nowshow that (Sl) holds. First, using (Sr), (Fl), and (Sr) again, we have((xy)/z)e(x) · z2 = xy · e(x)z = x · yz = (x/z)y · z2.Cancelling and dividing on the right by e(x), we obtain(*) (xy)/z = ((x/z)y)/e(x).Next we use (Sr), (Er), and (Sr) again to compute((xy)/e(z))z · e(z)2 = xy · z = xz · ye(z) = ((xz)/e(z))y · e(z)2.Cancelling, we obtain(**) ((xy)/e(z))z = ((xz)/e(z))y.Finally, we verify (Sl) as follows:xy · xz = ((xy)/z)x · z2 by (Sr)= [((x/z)y)/e(x)]x · z2 by (∗)= [((x/z)x)/e(x)]y · z2 by (∗∗)= (x2/z)y · z2 by (∗)= x2 · yz by (Sr).Axioms for trimedial quasigroups 293Since we have shown that Q is a semimedial, F-quasigroup, it follows from Propo-sition 1.1 that Q is trimedial.On the other hand, if Q is a right semimedial, left F-quasigroup, then Qopsatisfies (Sl) and (Fr) by Lemma 2.1(1), (2). By the preceding argument, Qop isa semimedial, F-quasigroup, and thus so is Q by Lemma 2.1(1), (2). Once again,Proposition 1.1 completes the proof. In closing, we note that further investigations suggest themselves. For example,it would be of interest to determine the structure of quasigroups that are onlyassumed to satisfy (El), or, in view of Lemma 2.5, those satisfying (El), (Sl) and(Fl). In this line we pose a couple of problems.Problem 2.8. 1. Characterize the loop isotopes of quasigroups satisfying (El).2. Characterize the loop isotopes of quasigroups satisfying (El), (Sl), and(Fl).References[1] Belousov V.D., Foundations of the Theory of Quasigroups and Loops, Izdat. Nauka,Moscow, 1967 (Russian); MR 36#1569, Zbl 163:01801.[2] Belousov V.D., About one quasigroup class (Russian), Uchenye zapiski Beltskogo gospedin-stituta im. A. Russo 5 (1960), 29–44.[3] Belousov V.D., Elements of Quasigroup Theory: A Special Course (in Russian), KishinevState University Press, Kishinev, 1981.[4] Bruck R.H., Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944),19–52; MR 5,229d, Zbl 0063.00635.[5] Bruck R.H., A Survey of Binary Systems, Springer-Verlag, 1971; MR 20#76, Zbl 206:30301.[6] Chein O., Pflugfelder H.O., Smith J.D.H. (eds.), Quasigroups and Loops: Theory and Ap-plications, Sigma Series in Pure Math. 9, Heldermann Verlag, 1990; MR 93g:21033, Zbl0719.20036.[7] Kepka T., Structure of triabelian quasigroups, Comment. Math. Univ. Carolinae 17 (1976),229–240; MR 53#10965, Zbl 0338.20097.[8] Kepka T., A note on WA-quasigroups, Acta Univ. Carolin. Math. Phys. 19 (1978), no. 2,61–62; MR 80b:20094, Zbl 0382.20057.[9] Kepka T., F-quasigroups isotopic to Moufang loops, Czechoslovak Math. J. 29 (1979),62–83; MR 80b:20095, Zbl 0444.20067.[10] Kepka T., Kinyon M.K., Phillips, The structure of F-quasigroups, in preparation.[11] Kinyon M.K., Phillips J.D., A note on trimedial quasigroups, Quasigroups & Related Sys-tems 9 (2002), 65–66; CMP 1 943 753, Zbl pre01894750.[12] McCune W.W., OTTER 3.3 Reference Manual and Guide, Technical MemorandumANL/MCS-TM-263, Argonne National Laboratory, 2003;or see http://www.mcs.anl.gov/AR/otter/.[13] Murdoch D.C., Quasi-groups which satisfy certain generalized associative laws, Amer. J.Math. 61 (1939), 509–522; Zbl 0020.34702.294 M.K.Kinyon, J.D.Phillips[14] Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Math. 8,Heldermann Verlag, Berlin, 1990; MR 93g:20132, Zbl 719:20036.[15] Toyoda K., On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221–227;MR 7,241g, Zbl 0061.02403.Department of Mathematical Sciences, Indiana University South Bend,South Bend, IN 46634, USAE-mail : mkinyon@iusb.eduhttp://mypages.iusb.edu/~mkinyonDepartment of Mathematics & Computer Science, Wabash College,Crawfordsville, IN 47933, USAE-mail : phillipj@wabash.eduhttp://www.wabash.edu/depart/math/faculty.html#Phillips(Received November 19, 2003, revised January 30, 2004)

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Axioms for trimedial quasigroups

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