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On Birkhoff's Quasigroup Axioms On Birkhoff's Quasigroup Axioms J.D. Phillips∗,a, D.I. Pushkashub, A.V. Shcherbacovc, V.A. Shcherbacovb aNorthern Michigan University, Department of Mathematics and Computer Science, Marquette, Michigan bInstitute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei str. 5, MD-2028, Chi¸sin˘au,Moldova cTheoretical Lyceum \C.Sibirschi," Lech Kaczyski str. 4, MD-2028, Chi¸sin˘au,Moldova Abstract Birkhoff defined a quasigroup as an algebra (Q; ·; n; =) that satisfies the fol- lowing six identities: x · (xny) = y,(y=x) · x = y, xn(x · y) = y,(y · x)=x = y, x=(ynx) = y, and (x=y)nx = y . We investigate triples and tetrads of identi- ties composed of these six, emphasizing those that axiomatize the variety of quasigroups. Key words: (left) quasigroup, (equational) quasigroup, division groupoid, cancellation groupoid 2000 MSC: 20N05 1 1. Introduction and Terminology 2 A binary groupoid,(G; A), is a non-empty set G, together with a binary 3 operation A. It is customary to omit the adjective \binary" and to refer to 4 these simply as \groupoids." We will include the adjective \binary" in those 5 instances where we wish to emphasize it. 6 Moufang [9, 18] defined a quasigroup as a groupoid (Q; ◦) in which, for 7 all a; b 2 Q, there exist unique solutions x; y 2 Q to the equations x ◦ a = 8 b and a ◦ y = b. Moufang also included two of the laws that eventually 9 came to be called \the Moufang laws" in her definition of quasigroup, viz. 10 (x·y)·(z ·x) = x·((y ·z)·x). The definition of \quasigroup" has evolved a bit ∗Corresponding author Email addresses: [email protected] (J.D. Phillips), [email protected] (D.I. Pushkashu), [email protected] (A.V. Shcherbacov), [email protected] (V.A. Shcherbacov) Preprint submitted to Journal of Algebra February 26, 2016 11 over the years, so that today the word \quasigroup" refers to objects slightly 12 more structured than Moufang's, as we shall see, but that don't necessarily 13 satisfy the Moufang laws. Fuller accounts of basic terms and concepts, as well 14 as more comprehensive historical overviews, can be found in [2, 5, 13, 14, 20]. 15 Bates and Kiokemeister [1] noticed that the class of quasigroups defined 16 qua Moufang is not closed with respect to the taking of homomorphic images. 17 Explicitly, homomorphic images of quasigroups (defined in this manner) are 18 only division groupoids (definition given below). 19 This algebraic infelicity was fixed by Birkhoff who defined a quasigroup 20 using three binary operations and the six identities given above [6, 7]. Shortly 21 thereafter, T. Evans [11] shortened Birkhoff's list of six identities to four (we 22 give them, below). It follows, then, from the classic universal algebra theorem 23 named for Birkhoff, that the class of (equational) quasigroups is a variety and 24 is thus closed with respect to the taking of homomorphic images [7, 8]. In 25 this paper we give two triples of identities from Birkhoff's list of six, as well 26 as nine tetrads, each of which axiomatizes the variety of quasigroups. 27 Let (Q; ·) be a groupoid. We use the standard notation Lax = a · x for 28 all x 2 Q, to denote left translation by the fixed element a 2 Q. Right 29 translations are defined analogously. We define middle translations, Pa, as 30 follows: x·Pax = a for all x 2 Q [3]. The definition of middle ternary relation 31 for groupoids (an analogue of middle quasigroup translation) is given in [22]. 32 Finally, we recall another definition of \quasigroup" which is equivalent to 33 the equational definition given above: a binary groupoid (Q; A) with binary 34 operation A is called a binary quasigroup if, in the equality A(x1; x2) = x3, 35 knowledge of any two of x1; x2, and x3 specifies the third uniquely [4]. 36 From this definition it follows that with a given binary quasigroup (Q; A), 37 it is possible to associate (3!−1) others, the so-called parastrophes of (Q; A)| 38 there are six of them, as we shall see : 1:A(x1; x2) = x3 () (12) 2:A (x2; x1) = x3 () (13) 3:A (x3; x2) = x1 () (23) 4:A (x1; x3) = x2 () (123) 5:A (x2; x3) = x1 () (132) 6:A (x3; x1) = x2 39 [28, p. 230], [2, p. 18]. (12) (13) 40 Usually the operation A is denoted as ∗, the operation A is denoted 2 (23) (123) 41 as =, the operation A is denoted as n, the operation A denoted as ==, (132) 42 and the operation A denoted as nn. 43 The following table shows, for each kind of translation, the equivalent 44 one in each of the (six) parastrophes of a quasigroup (Q; ·). For example, in (23) · 45 Table 1 we have R = P . In fact, Table 1 is a rewritten form of results on " 46 three kinds of translations from [3]. See also [10, 22]. In Table 1 , A = A Table 1: Translations of quasigroup parastrophes. " (12) (13) (23) (123) (132) R R L R−1 P P −1 L−1 L L R P −1 L−1 R−1 P P P P −1 L−1 R L R−1 R−1 R−1 L−1 R P −1 P L L−1 L−1 R−1 P L R P −1 P −1 P −1 P L R−1 L−1 R 47 A groupoid (G; ·) is called a left cancellation groupoid if the translation 48 La is an injective map for every a 2 G. Similarly, a groupoid (G; ·) is called a 49 right cancellation groupoid if the translation Ra is an injective map for every 50 a 2 G. Finally, a groupoid (G; ·) is called a cancellation groupoid if it is both 51 a left and a right cancellation groupoid. 52 Example 1. Let x ◦ y = 2x + 3y for all x; y 2 Z, where (Z; +; ·) is the ring 53 of integers. It is easy to check that (Z; ◦) is a cancellation groupoid. 54 A groupoid (G; ·) is said to be a left division groupoid if Lx is surjective 55 for every x 2 G. Right disvision groups are defined analogously. A groupoid 56 (G; ·) is said to be a division groupoid if it is both a left and a right division 57 groupoid. 2 3 58 Example 2. Let x ◦ y = x + y for all x; y 2 C, where (C; +; ·) is the field 59 of complex numbers. It is easy to check that (C; ◦) is a division groupoid. 60 A left cancellation left division groupoid (Q; ◦) is called a left quasigroup. 61 Right quasigroups are defined analogously. A groupoid that is both a left 62 and a right quasigroup is called a quasigroup. 3 63 Some other conditions which guarantee that a binary groupoid is a quasi- 64 group are given in [8, 15, 16, 19, 22]. 65 Birkhoff [6, 7] defined an equational quasigroup as an algebra with three 66 binary operations (Q; ·; =; n) that satisfies the following six identities: x · (xny) = y (1) 67 (y=x) · x = y (2) 68 xn(x · y) = y (3) 69 (y · x)=x = y (4) 70 x=(ynx) = y (5) 71 (x=y)nx = y: (6) 72 Using the language of translations, identities (1){(6) can be written in 73 the following form: n LxLx = " (7) 74 = RxRx = " (8) 75 n LxLx = " (9) 76 = RxRx = " (10) 77 = n LxRx = " (11) 78 n = RxLx = ": (12) 79 Lemma 1. Let µ and ν be mappings on a non-empty set Q. If the product 80 µν is bijective, then the map µ is surjective, and the map ν is injective. −1 81 Proof. If µν = α, where α is a bijection, then (α µ)ν = 1Q. Then, by −1 82 [12, Proposition 1] the map α µ is surjective and the map ν is injective. −1 83 Therefore, the map µ is surjective, since the map α is bijective. 84 From the translation representation of identities (1){(6) and Lemma 1 85 it is clear why, in [27], the identities (1){(4) are called respectively (SL), 86 (SR), (IL), (IR). Explicitly, these identities guarantee that the left (L) and 87 right (R) translations of an algebra (Q; ·; =; n) relative to the operation\·" are 88 surjective (S) or injective (I) mappings of the set Q. 89 Accordingly, we can use (SP) to denote identity (5) and (IP) to denote (6) 90 since these identities guarantee that middle translations (P) of a quasigroup 4 91 (Q; ·; =; n) are respectively surjective and injective mappings relative to the 92 operation \·" [22]. 93 Notice, there are other identities which can be used (by the definition of 94 equational quasigroup): (x==y) · x = y (13) x==(y · x) = y (14) x · (ynnx) = y (15) (x · y)nnx = y (16) 95 where x1 · x2 = x3 , x2==x3 = x1 , x3nnx1 = x2 [22, 26]. 96 The following lemma is well known. 97 Lemma 2. In an algebra (Q; ·; n; =) satisfying identities (1) and (4), (5) also 98 holds [21, 22, 27]. 99 Proof. From identity (4) we have (x · (xny))=(xny) = x: (17) 100 But from identity (1), we have x · (xny) = y. Thus, from (17) we obtain 101 y=(xny) = x, i.e., we obtain (up to renaming of variables) (5). 102 Lemma 3. In an algebra (Q; ·; n; =) satisfying identities (3) and (2), (6) also 103 holds [21, 22, 27]. 104 Proof. We can re-write identity (3) in the following form: (x=y)n((x=y) · y) = y: (18) 105 By identity (2) we have (x=y) · y = x.
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