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, ·, \,/) that satisfies the fol- lowing six identities: x · (x\y) = y,(y/x) · x = y, x\(x · y) = y,(y · x)/x = y, x/(y\x) = y, and (x/y)\x = 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 ∈ Q, there exist unique solutions x, y ∈ 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 ∈ Q, to denote left translation by the fixed element a ∈ Q. Right 29 translations are defined analogously. We define middle translations, Pa, as 30 follows: x·Pax = a for all x ∈ 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 \, the operation A denoted as //, (132) 42 and the operation A denoted as \\.
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 ε 46 three kinds of translations from [3]. See also [10, 22]. In Table 1 , A = A
Table 1: Translations of quasigroup parastrophes.
ε (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 ∈ G. Similarly, a groupoid (G, ·) is called a 49 right cancellation groupoid if the translation Ra is an injective map for every 50 a ∈ 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 ∈ 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 ∈ 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 ∈ 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, ·, /, \) that satisfies the following six identities:
x · (x\y) = y (1)
67 (y/x) · x = y (2) 68 x\(x · y) = y (3) 69 (y · x)/x = y (4) 70 x/(y\x) = y (5) 71 (x/y)\x = y. (6)
72 Using the language of translations, identities (1)–(6) can be written in
73 the following form: \ LxLx = ε (7) 74 / RxRx = ε (8) 75 \ LxLx = ε (9) 76 / RxRx = ε (10) 77 / \ LxRx = ε (11) 78 \ / RxLx = ε. (12)
79 Lemma 1. Let µ and ν be mappings on a non-empty set Q. If the product
80 µν is bijective, then the map µ is surjective, and the map ν is injective.
−1 81 Proof. If µν = α, where α is a bijection, then (α µ)ν = 1Q. Then, by −1 82 [12, Proposition 1] the map α µ is surjective and the map ν is injective. −1 83 Therefore, the map µ is surjective, since the map α is bijective.
84 From the translation representation of identities (1)–(6) and Lemma 1
85 it is clear why, in [27], the identities (1)–(4) are called respectively (SL),
86 (SR), (IL), (IR). Explicitly, these identities guarantee that the left (L) and
87 right (R) translations of an algebra (Q, ·, /, \) 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, ·, /, \) 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 · (y\\x) = y (15) (x · y)\\x = y (16)
95 where x1 · x2 = x3 ⇔ x2//x3 = x1 ⇔ x3\\x1 = x2 [22, 26]. 96 The following lemma is well known.
97 Lemma 2. In an algebra (Q, ·, \,/) satisfying identities (1) and (4), (5) also
98 holds [21, 22, 27].
99 Proof. From identity (4) we have
(x · (x\y))/(x\y) = x. (17)
100 But from identity (1), we have x · (x\y) = y. Thus, from (17) we obtain
101 y/(x\y) = x, i.e., we obtain (up to renaming of variables) (5).
102 Lemma 3. In an algebra (Q, ·, \,/) 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)\((x/y) · y) = y. (18)
105 By identity (2) we have (x/y) · y = x. Thus, from (18) we have (x/y)\x = y,
106 i.e. we obtain (6).
107 Evans’ standard equational definition of a quasigroup (namely, that an
108 algebra (Q, ·, \,/) satisfying identities (1)–(4) is called a quasigroup [2, 5, 7,
109 8, 11, 20, 27]) then follows from Lemmas 2 and 3 .
110 The equivalence of Moufang’s definition and Evans’ definition is well
111 known [2, 21, 27]. In [21, p. 11] the following problem is posed: “Re-
112 search properties of an algebra (Q, ·, /, \) satisfying various combinations of
113 the identities (1)–(6)”. This investigation was begun in [22, 26]; the following
114 “equivalence” theorem is proved in [22].
5 115 Theorem 1. 1. A groupoid (Q, ·) is a left division groupoid if and only if
116 there exists a left cancellation groupoid (Q, \) such that in the algebra
117 (Q, ·, \), the identity (1) is satisfied.
118 2. A groupoid (Q, ·) is a right division groupoid if and only if there exists
119 a right cancellation groupoid (Q, /) such that in the algebra (Q, ·,/) ,the
120 identity (2) is satisfied.
121 3. A groupoid (Q, ·) is a left cancellation groupoid if and only if there
122 exists a left division groupoid (Q, \) such that in the algebra (Q, ·, \),
123 the identity (3) is satisfied.
124 4. A groupoid (Q, ·) is a right cancellation groupoid if and only if there
125 exists a right division groupoid (Q, /) such that in the algebra (Q, ·,/),
126 the identity (4) is satisfied.
127 From Theorem 1 it is possible to derive information regarding the proper-
128 ties of algebras that satisfy various combinations of identities that are formed
129 from the identities (1)–(4). For instance, if in (Q, ·, /, \) the identities (1),
130 (2), (3) are satisfied, then the groupoid (Q, ·) is a left quasigroup with right
131 division. Therefore, we can call an algebra (Q, ·, /, \) satisfying identities (1),
132 (2), (3) a left quasigroup with right division. In Example 5, such an object is
133 constructed.
134 2. Results
135 Lemma 4. In an algebra (Q, ·, \,/) satisfying identities (4) and (6), (3) also
136 holds.
137 Proof. From identity (6), and using → as a substitution symbol, (x → (x ·
138 y), y → x) it follows that ((x · y)/y)\(x · y) = y. (19)
139 By (4), we have (x · y)/y = x. Therefore, (19) takes the form x\(x · y) = y,
140 and it coincides with (3).
141 Lemma 5. In an algebra (Q, ·, \,/) satisfying identities (2) and (5), (1) also
142 holds.
143 Proof. In identity (2), change the variable y by the variable x, and the vari-
144 able x by the symbol (term) y\x; thus obtaining: (x/(y\x)) · (y\x) = x. (20)
6 145 By identity (5), we have x/(y\x) = y. Therefore, we can rewrite (20) in the
146 following form: y · (y\x) = x. (21)
147 Thus, we obtain (1).
148 Lemma 6. In an algebra (Q, ·, \,/) satisfying identities (1) and (6), (2) also
149 holds.
150 Proof. We can re-write identity (1) (x → (x/y), y → x) in the following form:
(x/y) · ((x/y)\x) = x. (22)
151 By identity (6), we have (x/y)\x = y. Therefore, (22) takes the form (x/
152 y) · y = x, and we have (2).
153 Lemma 7. In an algebra (Q, ·, \,/) satisfying the identities (3) and (5), (4)
154 also holds.
155 Proof. From identity (5) (x → (x · y), y → x) we have
(x · y)/(x\(x · y)) = x. (23)
156 By identity (3), we have x\(x · y) = y. Therefore, (23) takes the form (x · y)/
157 y = x, and we have (4).
158 We summarize the results from Lemmas 2–7 in the following Lemma.
159 Lemma 8. In an algebra (Q, ·, \,/) the following implications hold:
(1), (4) → (5), (3), (2) → (6), (4), (6) → (3), (2), (5) → (1), (24) (1), (6) → (2), (3), (5) → (4).
160 Theorem 2. An algebra (Q, ·, \,/) satisfying the identities (1), (4) and (6)
161 is a quasigroup [23–25].
162 Proof. This follows from Lemmas 4 and 6, or from the first column of impli-
163 cations of Lemma 8.
164 Theorem 3. An algebra (Q, ·, \,/) satisfying the identities (2), (3) and (5)
165 is a quasigroup [23–25].
7 166 Proof. This follows from Lemmas 5 and 7, or from the second column of
167 implications from Lemma 8.
168 Clearly, there are 15 tetrads of identities which can be composed from
169 Birkhoff’s six identities. In the following corollary we define, in eight dif-
170 ferent ways, the variety of equational quasigroups using four identities from
171 Birkhoff’s list of six.
172 Corollary 1. An algebra (Q, ·, \,/) satisfying any of the following eight
173 tetrads of identities is a quasigroup:
174 1. (1), (2), (3) and (5)
175 2. (2), (3), (4) and (5)
176 3. (1), (2), (4) and (6)
177 4. (1), (3), (4) and (6)
178 5. (1), (3), (5) and (6)
179 6. (1), (4), (5) and (6)
180 7. (2), (3), (5) and (6)
181 8. (2), (4), (5) and (6).
182 Proof. Case 1 follows from Theorem 3; Case 2, from Theorem 3; Case 3, from
183 Theorem 2; Case 4, from Theorem 2; Case 5, from Lemmas 6 and 7; Case
184 6, from Theorem 2; Case 7, from Theorem 3; Case 8, from Lemmas 5 and
185 4.
186 Corollary 2. An algebra (Q, ·, \,/) satisfying any five of the six Brikhoff
187 identities is a quasigroup.
188 Proof. This follows from Lemma 2 and Corollary 1.
189 There are thus nine tetrads (the eight from the corollary, plus the stan-
190 dard tetrad given by (1), (2), (3), (4)), each of which defines the variety of
191 quasigroups.
192 3. Some non-quasigroup tetrads
193 We give examples which demonstrate that some combinations of four
194 identities from Birkhoff’s six do not define equational quasigroups.
8 195 Example 3. Define a binary groupoid (Z, ∗) by x∗y = x+2y for all x, y ∈ Z, 196 where (Z, +, ·) is the ring of integers. It is easy to check that the groupoid 197 (Z, ∗) is a right quasigroup with the left cancellative property. From Theorem 198 1 it follows that the groupoid (Z, ∗) satisfies the identities (2), (3), (4). 199 Next we prove that the groupoid (Z, ∗) satisfies identity (6). If x ∗ y = 200 x + 2y, then the set of solutions of the equation a ∗ y = b is described thusly: b−a 201 y = a\b = 2 . Clearly, solutions of this equation exist only for any pair of 202 integers of equal parity.
203 Now, notice that solutions of the equation x ∗ a = b have the following
204 form: x = b/a = b − 2a. Thus,
a − a + 2b (a/b)\a = (a − 2b)\a = = b. (25) 2
205 Notice now that the left part of equality (25) contains the proper subterm 206 (a/b) which is defined for all a, b ∈ Z. The expression (a/b)\a is also defined 207 for all elements a, b ∈ Z. Therefore, the equality (a/b)\a = b is an identity in 208 the groupoid (Z, ∗). And so we see that the groupoid (Z, ∗) satisfies identities 209 (2), (3), (4), and (6) but is not a quasigroup.
210 Example 4. Define a binary groupoid (Z, ∗) by x ∗ y = x + [y/2] for all 211 x, y ∈ Z, where (Z, +, ·) is the ring of integers; [y/2] = a, if y = 2 · a; and 212 [y/2] = a, if y = 2 · a + 1.
213 It is easy to check that this groupoid is a right quasigroup with left divi- 214 sion. From Theorem 1 it follows that the groupoid (Z, ∗) satisfies identities 215 (1), (2) and (4). 216 Next, we show that the groupoid (Z, ∗) satisfies identity (5). If x ∗ y = 217 x + [y/2], then the set of solutions of the equation b ∗ y = a is described
218 thusly: y = −2b + 2a + i, where i = 0, 1. As usual, denote this set by b\a. Similarly, solutions of the equation x ∗ b = a have the following form: x = a − [b/2] and are denoted by a/b. Then
−2b + 2a + i a/(b\a) = a/(−2b + 2a + i) = a − = 2 i i a − −b + a + = a + b − a − = b. 2 2
219 Here, we have used the fact that the elements a and b are integers and that i 220 2 = 0 since i = 0, 1.
9 221 Therefore, a/(b\a) = b for all a, b ∈ Z, and the groupoid (Z, ∗) satisfies 222 identities (1), (2), (4), and (5) but is not a quasigroup.
223 Example 5. Define a binary groupoid (Z, ∗) by x ∗ y = [x/2] + y for all 224 x, y ∈ Z, where (Z, +, ·) is the ring of integers, and [x/2] = a, if x = 2 · a; 225 [x/2] = a, if x = 2 · a + 1. 226 It is easy to check that (Z, ∗) is a left quasigroup with right division. 227 From Theorem 1 it follows that (Z, ∗) satisfies the identities (1), (2), and (3). 228 As in Example 4, one can show that (Z, ∗) satisfies the identity (6). There- 229 fore, (Z, ∗) satisfies the identities (1), (2), (3), and (6) but is not a quasigroup.
230 Example 6. Define a binary groupoid (Z, ∗) by x∗y = 2x+y for all x, y ∈ Z, 231 where (Z, +, ·) is the ring of integers. It is easy to check that (Z, ∗) is a left 232 quasigroup with the right cancellation property. From Theorem 1 it follows 233 that (Z, ∗) satisfies identities (1), (3), and (4). 234 As in Example 3, one can show that (Z, ∗) satisfies identity (5). Therefore, 235 (a/b)\a = b for all a, b ∈ Z, and so (Z, ∗) satisfies identities (1), (3), (4), and 236 (5) but is not a quasigroup.
237 While none of the tetrads in these four examples axiomatizes the variety
238 of quasigroups, each imposes some structure, as the following theorem shows.
239 Theorem 4.
240 1. If an algebra (Q, ·, \,/) satisfies the identities (2), (3), (4), and (6),
241 then (Q, ·) is a right quasigroup with the left cancellative property.
242 2. If an algebra (Q, ·, \,/) satisfies the identities (1), (2), (4), and (5),
243 then (Q, ·) is a right quasigroup with left division.
244 3. If an algebra (Q, ·, \,/) satisfies the identities (1), (2), (3), and (6),
245 then (Q, ·) is a left quasigroup with right division.
246 4. If an algebra (Q, ·, \,/) satisfies the identities 1), (3), (4), and (5), then
247 (Q, ·) is a left quasigroup with the right cancellative property.
248 Proof. Case 1. From Theorem 1, it follows that if an algebra (Q, ·, \,/) satis-
249 fies identities (2), (3), and (4), then groupoid (Q, ·) is a right quasigroup with
250 the left cancellative property. From Lemma 3, it follows that (2) together
251 with (3) imply (6). Example 3 demonstrates that there exist non-quasigroup
252 algebras (Q, ·, \,/) satisfying this given set of identities.
253 Case 2. From Theorem 1, it follows that if an algebra (Q, ·, \,/) satisfies
254 the identities (1), (2), and (4), then (Q, ·) is a right quasigroup with the
10 255 left division property. From Lemma 2, it follows that (1) together with
256 (4) implies (5). Example 4 demonstrates that there exist non-quasigroup
257 algebras (Q, ·, \,/) satisfying this given set of identities.
258 Case 3. From Theorem 1, it follows that if an algebra (Q, ·, \,/) satisfies
259 the identities (1), (2), and (3), then (Q, ·) is a left quasigroup with right
260 division. From Lemma 3, it follows that (2), together with (3), imply (6).
261 Example 5 demonstrates that there exist non-quasigroup algebras (Q, ·, \,/)
262 satisfying this given set of identities.
263 Case 4. From Theorem 1, it follows that if an algebra (Q, ·, \,/) satisfies
264 the identities (1), (3), and (4), then (Q, ·) is a left quasigroup with the right
265 cancellative property. From Lemma 2 it follows that (1) together with (4)
266 imply (5). Example 6 demonstrates that there exist non-quasigroup algebras
267 (Q, ·, \,/) satisfying this given set of identities.
268 4. Summary and Open Problems
269 The next theorem deals with the two remaining tetrads.
270 Theorem 5.
271 1. If an algebra (Q, ·, \,/) satisfies the identities (1), (2), (5), and (6),
272 then (Q, ·) is a division groupoid such that the groupoid (Q, /) is a
273 right cancellative, left quasigroup, and the groupoid (Q, \) is a left can-
274 cellative, right quasigroup.
275 2. If an algebra (Q, ·, \,/) satisfies the identities (3), (4), (5), and (6),
276 then (Q, ·) is a cancellation groupoid such that the groupoid (Q, /) is a
277 left quasigroup with a right division, and the groupoid (Q, \) is a right
278 quasigroup with a left division.
279 Proof. Case 1. The fact that the groupoid (Q, ·) is division groupoid follows
280 from Theorem 1. The rest follows from the given identities and Lemma 1.
281 Case 2 is proved analogously.
282 In summary, then, we have shown that there are nine different tetrads of
283 identities composed from Birkhoff’s six, each of which axiomatizes the variety
284 of quasigroups. We have proven structural theorems about the varieties
285 axiomatized by the remaining six tetrads. And we have shown that four of
286 these remaining six do not axiomatize the variety of quasigroups. Thus, we
287 end this paper with the following pair of open problems.
11 288 Problem 1. Must an algebra (Q, ·, \,/) that satisfies identities (1), (2), (5),
289 and (6) be a quasigroup?
290 Problem 2. Must an algebra (Q, ·, \,/) that satisfies identities (3), (4), (5),
291 and (6) be a quasigroup?
292 5. Acknowledgements
293 Our investigations were aided by the automated deduction tool Prover9
294 [17].
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