PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 93 (107) (2013), 29–47 DOI: 10.2298/PIM1307029K

QUADRATIC LEVEL QUASIGROUP EQUATIONS WITH FOUR VARIABLES II: THE LATTICE OF VARIETIES

Aleksandar Krapež

Communicated by Siniša Crvenković

Abstract. We consider a class of quasigroup identities (with one operation symbol) of the form x_{1}x_{2} · x_{3}x_{4} = x_{5}x_{6} · x_{7}x_{8} and with x_{i} ∈ {x, y, u, v} (1 ⩽ i ⩽ 8) with each of x, y, u, v occurring exactly twice in the identity. There are 105 such identities. They generate 26 quasigroup varieties. The lattice of these varieties is given.

1. Introduction

In the previous paper Krapež [2] we defined the quadratic level quasigroup equations with four variables. They are quadratic equations of the form:

(L2)  x_{1}x_{2} · x_{3}x_{4} = x_{5}x_{6} · x_{7}x_{8}

where x_{i} ∈ {x, y, u, v} (1 ⩽ i ⩽ 8). The operation · is assumed to be a quasigroup. No division operation occurrs in the equation (L2). There are 105 such equations. The complete list is given in [2, equations (4.1)–(4.105)] where all definitions of undefined notions and further references can be found. The general solutions of these 105 equations are given in [2]. Since quasigroups are defined as models of identities (in the language {·, \, /}) and equations are also identities, the sets of solutions to above equations are quasigroup varieties. The equations combine into 19 classes of equivalent equations resulting in 19 quasigroup varieties:

(Q)  x = x  (Quasigroups)
(C)  xy = yx  (Commutative quasigroups)
(B11)  xy · uv = vu · yx (4–palindromic quasigroups)

2010 Mathematics Subject Classification: Primary 20N05; Secondary 08B15, 39B52.

Key words and phrases: quasigroup, quasigroup functional equation, quadratic level quasigroup equation, quasigroup identity, quasigroup variety, lattice of varieties.

Supported by Ministry of Education, Science and Technological Development of Serbia through projects ON 174008 and ON 174026.

(U1)  xy · yx = e  (Skew symmetric quasigroups)
(U)  xx = e  (Unipotent quasigroups)
(Ub0)  (U), (b0)  (Unipotent b0-quasigroups)
(Ub1)  (U), (b1)  (Unipotent b1-quasigroups)

(CU)

(U), (C)

(Commutative unipotent quasigroups)

(LLU)

(U), (LL)

(Unipotent left linear quasigroups)

(RLU)

(U), (RL)

(Unipotent right linear quasigroups)

(M)  xy · uv = xu · yv  (Medial quasigroups)
(P)  xy · uv = vy · ux  (Paramedial quasigroups)

(T1)

(C), (M)

(Commutative medial quasigroups)

(D1)  xy · xu = uv · yv
(I)  xy · yu = xv · vu  (Intermedial quasigroups)
(E)  xy · ux = vy · uv  (Extramedial quasigroups)
(ME)  xy · ux = vu · yv
(PI)  xy · yu = uv · vx
(BT1)  xy · xu = yv · uv

We assume:

ex = xe (b0)

e · xy = yx · e (b1)

x(u\y) · z = x(u\u) · (u\yz) (LL)

x · (y/u)z = (xy/u) · (u/u)z (RL)

Seven more varieties are defined by the systems of two identities (see Table 1).

\begin{matrix}U1B11 & (U1), (B11) \\ UB11 & (U), (B11) \\ T 11 & (M), (P) \\ D11 & (E), (I) \\ BM & (M), (I) \\ BP & (P), (E) \\ BT 11 & (M), (P I) \end{matrix}

Table 1. Varieties of quasigroups (two identities)

None of the systems is equivalent to just one identity with four variables. However, every one of the systems is equivalent to a single quadratic identity with eight variables. For example (T 11) is equivalent to (xy · uv)(pq · rs) = (xu · yv)(sq · rp).

We get more information on some of these varieties by looking at general solutions of the given (systems of) equations. By [2, Theorems 9,10]:

Theorem 1.1. Quasigroups (S; ·, \, /) which belong to the variety LLU (RLU) are representable as

xy = Ax − Ay + c (LLU)

xy = c − Ax + Ay (RLU)

where + is an arbitrary group on S, A is an automorphism of + and c is any element of S.

Similarly, the quasigroups from varieties in Table 2 are linear over an Abelian group + (i.e., xy = Ax + By + c; A, B–automorphisms of +) and satisfy further conditions depending on the particular variety (see Table 2).

variety identities conditions on +  conditions on A, B
M  (M)  Abelian group  AB = BA
P  (P)  Abelian group  A^{2} = B^{2}
E  (E)  Abelian group  A^{2} + B^{2} = O
I  (I)  Abelian group  AB + BA = O
ME  (ME)  Abelian group AB = BA, A^{2} + B^{2} = O
PI  (P I)  Abelian group A^{2} = B^{2}, AB + BA = O
T11 (M), (P) Abelian group  AB = BA, A^{2} = B^{2}
D11  (E), (I)  Abelian group A^{2}+B^{2} =O, AB+BA=O
T1  (T 1)  Abelian group  A = B
D1  (D1)  Abelian group  A + B = O
BM  (M), (I) Boolean group  AB = BA
BP  (P), (E) Boolean group  A^{2} = B^{2}
BT11 (M), (P I) Boolean group  A^{2} = B^{2}, AB = BA
BT1  (BT 1)  Boolean group  A = B

Table 2. Representation of quasigroups from T–quasigroup varieties in Q_{4}

The data from Table 2, suggest relationship between varieties of Abelian group isotopes given in Figure 1 (with variety Q added).

In such graphs it is customary that two nodes V and W (W above V) connected by the line represent a relationship V ◁ W of W being immediately above V. The relation < is the transitive closure of ◁. No connection between V and W means that V and W are incomparable (denoted V ∥W). We informally say that the graph in Figure 1 is valid in the strong sense. At the moment we are far from proving such strong relationship between nodes of the graph in Figure 1. All we can say now is that V ⊆ W for V and W connected by a line (with W above V), while not having a line connecting V and W does not necessarily mean V ∥W. Therefore the graph in Figure 1 is valid in the weak sense only. Assumption is similar for Figures 2 and 3.

BT 1 D1  BT 11  T 1 BM  P I D11 EM  T 11  BP P I  E M  Q

Figure 1. Varieties of quasigroups of Abelian group isotopes (with Q added)

The lattice of varieties of quasigroups which are not necessarily group isotopes is given in Figure 2. However, we have to justify relationships (even the weak ones) between varieties in this case.

Lemma 1.1. The following relationships hold between varieties of quasigroups which are not necessarily group isotopes.

1. CU ⊆ C ; 2. CU ⊆ UB11 ; 3. CU ⊆ U1B11 ; 4. CU ⊆ Ub0 ; 5. C ⊆ B11 ; 6. UB11 ⊆ Ub1 ; 7. UB11 ⊆ B11  8. U1B11 ⊆ U1 ; 9. U1B11 ⊆ B11 ; 10. Ub0 ⊆ U ; 11. Ub1 ⊆ U ; 12. B11 ⊆ Q ; 13. U1 ⊆ Q ; 14. U ⊆ Q.

Proof. 1. By the definition of CU.

2. Assume (C). Then xy · uv = yx · vu = vu · yx.

3. Assume (C) and (U). Then (B11) follows by 2. Also xy · yx = xy · xy = e i.e. (U1).

4. (Ub0) is a special case of (CU).

CU U1B11  UB11 C Ub0  Ub1 U1  U  B11 Q

Figure 2. Varieties of quasigroups which are not necessarily group isotopes

5. As in 2.

6. Assume (B11). Then e · xy = zz · xy = yx · zz = yx · e. 7–14. Trivial. □

There are two varieties which are not included in graphs in Figures 1 and 2: LLU and RLU. They are elements of the subset {LLU, LRU, Ub1, D1} with order relations as indicated in Figure 3. We have to justify this claim also.

D1 LLU  RLU Ub1

Figure 3. An ordered subset of Q_{4} containing LLU and RLU

By Theorem 1.1, an operation · in a left linear unipotent quasigroup is of the form x · y = Ax − Ay + c for some group +, automorphism A and an element c. By simple checking we prove (LLU) ⇒ (Ub1). Similarly, using entry for D1 in Table 2, we prove (D1) ⇒ (LLU). Therefore, D1 ⊆ LLU ⊆ Ub1. By the left–right duality principle for groupoids, we have D1 ⊆ RLU ⊆ Ub1 as well.

Q U1  U  B11  I  E  P  M Ub0  Ub1  D11  P I  T 11  BM  ME  BP U1B11  UB11  C  LLU  RLU CU  T 1  D1  BT 11 BT 1

Figure 4. The lattice of all varieties from Q_{4} There are several more relationships with which we have to deal separately.

Lemma 1.2. The following relationships hold between the indicated varieties.

1. T 11 ⊆ B11 ; 2. D11 ⊆ U1 ; 3. BT 11 ⊆ U1B11 ; 4. T 1 ⊆ C ; 5. D1 ⊆ UB11  6. BT 1 ⊆ CU ; 7. M ⊆ Q ; 8. P ⊆ Q ; 9. E ⊆ Q ; 10. I ⊆ Q

The proof of 1 to 6 uses appropriate entries from Table 2 and requires simple checking only. Relations 7 to 10 are obvious.

Our ultimate goal is to prove that the ordered set^{1} (Q_{4}; ⊆) of 26 varieties of quasigroups axiomatized by one or more of 105 quadratic level identities is the lattice given in Figure 4.

2. The partition of Q_{4}

In order to make the proof of the stated result easier, we divide the set Q_{4} into sets of varieties of quasigroups which consist of:

• Boolean group isotopes: BT 1, BT 11, BM, BP

• Abelian group isotopes but not neccessarily Boolean group isotopes: M, P, E, I, ME, P I, T 11, D11, T 1, D1

• Group isotopes but not neccessarily Abelian group isotopes: LLU, RLU • Not neccessarily group isotopes: Q, C, B11, U1, U1B11, U, Ub0, Ub1,

UB11, CU.

The partition is based on the results of Sections 5–9 of [2], but we should emphasize again that the 26 varieties above are not yet proven to be different one from the other. However, by the well known result of quasigroup theory, that if a loop is isotopic to a group, then they are isomorphic, it follows that the four classes above are pairwise disjoint.

Another, independent partition of Q_{4} is:

• varieties with unipotent quasigroups only: U, Ub0, Ub1, UB11, CU, LLU, RLU, D1, BT 1

• varieties which contain quasigroups which are not neccessarily unipotent: Q, C, B11, U1, U1B11, M, P, E, I, ME, P I, T 11, D11, T 1, BM, BP, BT 11.

The above partition is justified by:

Lemma 2.1. All the quasigroups from varieties U, Ub0, Ub1, UB11, CU, LLU, RLU, D1, BT 1 are unipotent.

Every one of the varieties Q, C, B11, U1, U1B11, M, P, E, I, ME, P I, T 11, D11, T 1, BM, BP, BT 11 contains a quasigroup which is not unipotent.

Proof. The proof of the first part of the lemma follows from [2, Theorems 6.1–6.4, 7.1, 7.2, 8.3, 9.1] and the definition of (UB11). The second part is from the following two examples. □

1To avoid foundational issues, we work within a given universal set.

Example 2.1. Let (ℂ_{2}^{4}; +) be the fourth power of the two–element (Boolean) group (ℂ_{2}; +). For

A = [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] and B = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ]

define operation ⊕ on ℂ_{2}^{4} by x⊕y = Ax+By. We can easily see that A^{2} = B^{2} = Id as well as AB = BA and therefore (ℂ_{2}^{4}; ⊕) is a model of (BT 11) (also of (Q), (U1), (B11), (U1B11), (M), (P), (E), (I), (ME), (P I), (T 11), (D11), (BM) and (BP)) but not of (U).

Example 2.2. Let (ℝ; +) be the additive group of reals. Then it is an Abelian group isotope (A = B = 1) and therefore a model of (T 1) and (C), but because x + x = 2x ≠ 0 it is not a model of (U).

The meet of these two partitions is a partition related to an equivalence on Q_{4} which we denote by ∼.

Definition 2.1. The system (Q_{4}/∼ ; ⩽) is defined by:

\begin{aligned}𝒬 & = & \{Q, C, B11, U1, U1B11\} \\ 𝒰 & = & \{U, Ub0, Ub1, UB11, CU\} \\ 𝒢 & = & \{LLU, RLU\} \\ 𝒜 & = & \{M, P, E, I, ME, P I, T 11, D11, T 1\} \\ 𝒟 & = & \{D1\} \\ ℬ & = & \{BM, BP, BT 11\} \\ 𝒵 & = & \{BT 1\} \\ 𝒱 ⩽ 𝒲 & iff & ⋃ V ⊆ ⋃ W. \end{aligned}

The lattice of these classes is given in Figure 5.

𝒵 ℬ 𝒜 𝒬  𝒟 𝒢 𝒰

Figure 5. The lattice of ∼–classes in Q_{4}

Theorem 2.1. The function f : Q_{4} −→ Q_{4}/∼ (f(V) = V^{∼} ) is an order

preserving surjection but is not a lattice homomorphism.

Proof. The first part of the statement is obvious (check Figures 4 and 5). To see that f is not a homomorphism take M and I. Then f(M ∩ I) = f(BM) = ℬ while f(M) ∧ f(I) = 𝒜 ∧ 𝒜 = 𝒜. □

If V, W are varieties from classes 𝒱, 𝒲 respectively, in general there are four possibilities for the relationship between V and W: V = W, V ⊂ W, V ⊃ W and V ∥ W. However, if 𝒱 < 𝒲, then only two possibilities remain: either V ⊂ W or V ∥W. This is the reason for the introduction of the equivalence ∼ and its classes. For the reference, we formulate the above and two similar results as a separate Lemma and use it extensively in the rest of the paper.

Lemma 2.2. Let V, W be varieties from classes 𝒱, 𝒲 respectively.

• If 𝒱 < 𝒲 then either V ⊂ W or V ∥W.

• If V ∧ W ∉ 𝒱 ∪ 𝒲 then V ∥W.

• In particular, if 𝒱∥𝒲 then V ∥W.

3. The main result

Some parts of Q_{4} are either well known or trivial. For example, the variety Q is the greatest and the variety BT 1 is the smallest element. The first fact is obvious. The second fact follows from the easily verifiable property of any unipotent quasigroup, linear over a Boolean group, that it satisfies all 105 quadratic level equations with four variables. Namely, every quasigroup from BT 1 is of the form xy = Ax + Ay + c, where + is a Boolean group, A is an automorphism of + and c ∈ S. By [2, Lemma 8.1], any equation (L2) reduces to AA(x_{1} + ⋯+ x_{4}) = AA(x_{5} + ⋯+ x_{8}). This is equivalent to ∑ _{i=1}^{8} x_{i} = 0 which is always true since every variable appears exactly twice in the sum. Therefore:

Lemma 3.1. For every variety V from Q_{4} we have BT 1 ⊆ V.

But we want to prove that BT 1 ≠ V for any variety V from Q_{4} (except BT 1 itself). This is obvious as the variety BT 1 is the single element in the class 𝒵.

Another part of the lattice (Q_{4}; ⊆) that is known, is the lattice of all varieties of quasigroups defined by balanced identities with four variables, given implicitly in Förg–Rob, Krapež [1] and reproduced from Krapež [2] as Figure 6 here.

We proceed by proving the rest of the relationships among varieties from Q_{4}. Because of 𝒟∥ℬ, we have:

Lemma 3.2. The variety D1 is incomparable to all varieties from ℬ i.e.,

\begin{aligned}& BT11 & BM & BP \\ D1 & ∥ & ∥ & ∥ \end{aligned}

If we force the requirement that + is Boolean on (M), (P), (T 11), we get (BM), (BP), (BT 11) respectively (see Table 2). The order ⊆ on corresponding varieties is inherited, but we need to prove that BM, BP, BT 11 are all different

T 1 C  T 11 B11  M  P Q

Figure 6. Varieties of quasigroups defined by balanced identities

one to the other. The following examples prove that BT 11 ⊂ BM, BT 11 ⊂ BP and BM∥BP.

Example 3.1. Let 𝕍 = (V; +) with V = {0, 1, 2, 3} be a four-group, let A = (123), B = (132) and let (V; ⊕) be a quasigroup defined by x ⊕ y = Ax + By. Then, because AB = Id = BA and A^{2} = B ≠ A = B^{2}, the quasigroup (V; ⊕) is a model of (BM) but of neither (BT 11) nor (BP). This proves BT 11 ⊂ BM and BM ⊈ BP.

Example 3.2. Let 𝕍 be as in Example 3.1, let A = (12), B = (13) and let (V; ⊕) be defined by x ⊕ y = Ax + By. Then, because A^{2} = Id = B^{2} and AB = (123) ≠ (132) = BA, the quasigroup (V; ⊕) is a model of (BP) but not of (BM), which proves both BT 11 ⊂ BP and BP ⊈ BM.

Therefore we proved:

Lemma 3.3. The relationship between varieties from ℬ is given by the following table:

BT11 BM BP
BT11 =  ⊂ ⊂
BM  ⊃  =  ∥
BP  ⊃  ∥ =

We now give some examples which will be needed later.

Example 3.3. Let (ℝ; −) be the groupoid of reals under subtraction. Then (ℝ; −) is an Abelian group isotope (Ax = x, Bx = −x) and satisfies A + B = 0, A^{2} = 1 = B^{2}, AB = −1 = BA and therefore (ℝ; −) is a model of (D1), (P), (M) and (T 11). However, A = 1 ≠ −1 = B, A^{2} + B^{2} = 2 ≠ 0, AB + BA = −2 ≠ 0 which implies that (ℝ; −) is a model of neither (T 1), (D11), (ME), (E), (I) nor (P I).

Example 3.4. Let (ℝ^{2}; +) be the additive group of pairs of reals and x ⊕ y = Ax + By, where A = [\begin{matrix}0 & 3 \\ 3 & 0 \end{matrix} ] and B = [ \begin{matrix}4 & 5 \\ −5 & −4 \end{matrix} ]. Then (ℝ^{2}; ⊕) is an Abelian group isotope that satisfies A^{2} + B^{2} = O, AB + BA = O and therefore (ℝ^{2}; ⊕) is a model of (E), (I) and (D11). Also, A ≠ B, A+B ≠ O, A^{2} ≠ B^{2}, AB ≠ BA and therefore (ℝ^{2}; ⊕) is not a model of (T 1), (D1), (M), (P), (T 11), (ME), (P I).

Example 3.5. Let (ℂ; +) be the additive group of complex numbers and x⊕y = x+iy. Then (ℂ; ⊕) is an Abelian group isotope that satisfies A^{2}+B^{2} = 0, AB = BA and therefore (ℂ; ⊕) is a model of (M), (E) and (ME). Likewise, from A ≠ B, A + B ≠ 0, A^{2} ≠ B^{2}, AB + BA ≠ 0 it follows that (ℂ; ⊕) is not a model of (T 1), (D1), (I), (P), (T 11), (D11), (P I).

Example 3.6. Let (ℚ; +) be the additive group of quaternions and x⊕y = ix+ jy. Then (ℚ; ⊕) is an Abelian group isotope that satisfies A^{2} = B^{2}, AB + BA = 0 and therefore (ℚ; ⊕) is a model of (P), (I) and (P I). On the other hand A ≠ B, A + B ≠ 0, A^{2} + B^{2} ≠ 0, AB ≠ BA and therefore (ℚ; ⊕) is not a model of (T 1), (D1), (M), (E), (T 11), (D11), (ME).

Lemma 3.4. The relationship between varieties from 𝒟 and 𝒜 is given by the following table:

T1 T11 D11 ME PI M P E I ; D1 ∥ ⊂  ∥  ∥ ∥ ⊂ ⊂ ∥ ∥

Proof. 1. D1 ⊈ T 1 by Example 3.3. T 1 ⊈ D1 by Example 2.2.

2. We have D1 ⊆ T 11. Since D1 and T 11 belong to different classes 𝒟 and 𝒜, they are different too.

3. D1 ⊈ D11 by Example 3.3; D11 ⊈ D1 by Example 3.4.

4. D1 ⊈ ME by Example 3.3; ME ⊈ D1 by Example 3.5.

5. D1 ⊈ P I by Example 3.3; P I ⊈ D1 by Example 3.6.

6. D1 ⊂ M and D1 ⊂ P follow by the transitivity of ⊂.

7. D1 ⊈ E by Example 3.3; E ⊈ D1 by Example 3.5.

8. D1 ⊈ I by Example 3.3; I ⊈ D1 by Example 3.6. □

Lemma 3.5. The relationship between varieties from ℬ and 𝒜 is given by the following table:

T1 T11 D11 ME PI M P E I
BT11 ∥ ⊂  ⊂  ⊂ ⊂ ⊂ ⊂ ⊂ ⊂
BM ∥ ∥  ∥  ∥ ∥ ⊂ ∥ ∥ ⊂
BP ∥ ∥  ∥  ∥ ∥ ∥ ⊂ ⊂ ∥

Proof. 1. Since BT 11 consists of Boolean group isotopes and T 1 of T - quasigroups such that A = B, we conclude that BT 11 ∩ T 1 = BT 1. As BT 1 belongs to the class 𝒵, it is different from both BT 11 and T 1. Therefore BT 11∥T 1.

2. That BT 11 is strictly smaller than all other elements of 𝒜 follow from the fact that BT 11 does not belong to 𝒜.

3. Similarly, BM does not belong to 𝒜 and therefore BM ⊂ M and BM ⊂ I.

4. BM ∩ T 1 = BT 1 which belongs to 𝒵 and consequently BM∥T 1.

5. BM ∩ T 1 ⊆ BT 11 ∩ T 1 = BT 1 and BM∥T 1.

6. The meet of BM and any of T 11, D11, ME, P I, P, E is BT 11 and so BM is incomparable to any of them.

7. The proof for entries of BP is analogous to 3–6. □

If we force + to be Boolean on (I), (E), (D11) we get (BM), (BP), (BT 11) respectively. The order is preserved and since the later varieties are different, the mapping from I, E, D11 to BM, BP, BT 11 is surjective. Therefore:

Lemma 3.6. We have D11 ⊂ I, D11 ⊂ E, and I∥E.

The same schema we can apply to M, E, ME and conclude:

Lemma 3.7. The following relationships are true: ME ⊂ M, ME ⊂ E, and M∥E.

Again, applying the scheme to I, P, and P I we get:

Lemma 3.8. The relationships P I ⊂ I, P I ⊂ P, and I∥P hold.

Lemma 3.9. The relationship between varieties from 𝒜 is given by the following table:

T1 T11 D11 ME PI M P E I
T1 = ⊂  ∥  ∥ ∥ ⊂ ⊂ ∥ ∥
T11 ⊃ =  ∥  ∥ ∥ ⊂ ⊂ ∥ ∥
D11 ∥ ∥  =  ∥ ∥ ∥ ∥ ⊂ ⊂
ME ∥ ∥  ∥  = ∥ ⊂ ∥ ⊂ ∥
PI ∥ ∥  ∥  ∥ = ∥ ⊂ ∥ ⊂
M ⊃ ⊃  ∥  ⊃ ∥ = ∥ ∥ ∥
P ⊃ ⊃  ∥  ∥ ⊃ ∥ = ∥ ∥
E ∥ ∥  ⊃  ⊃ ∥ ∥ ∥ = ∥
I  ∥ ∥  ⊃  ∥ ⊃ ∥ ∥ ∥ =

Proof. 1. From 6 we see that T 1 ⊂ T 11, T 1 ⊂ M and T 1 ⊂ P.

2. By Example 2.2, (ℝ, +) is the model of (T 1) but none of: (D11), (ME), (P I), (E), (I). This proves that T 1 is not a subset of any of D11, ME, P I, E, I.

Following Example 3.4, (ℝ^{2}, ⊕) is a model of (D11), (E), (I) but not of (T 1). This proves T 1∥D11, T 1∥E and T 1∥I.

Following Example 3.5, (ℂ, ⊕) is a model of (ME) but not of (T 1). This proves ME ⊄ T 1 and consequently T 1∥ME.

Finally, following Example 3.6, (ℚ, ⊕) is the model of (P I) but not of (T 1), which proves P I ⊄ T 1 and therefore T 1∥P I.

3. Analogously, using the same models but with T 11 instead of T 1, we can prove incomparability of T 11 to all of (D11), (ME), (P I), (I), (E).

4. Following Lemma 3.6, we have D11 ⊂ E and D11 ⊂ I.

5. Following Example 3.4, (ℝ^{2}; ⊕) is a model of (D11) but of neither (ME) nor (P I), (M), (P).

Following Example 3.5, (ℂ; ⊕) is a model of (ME) and (M) but not of (D11). Therefore D11∥ME and D11∥M.

Following Example 3.6, (ℚ; ⊕) is a model of (P I) and (P) but not of (D11). Consequently, D11∥P I and D11∥P.

6. To prove that neither of ME, M, E is a subset of any of P I, P, I use Example 3.5. To prove that neither of P I, P, I is a subset of any of ME, M, E use Example 3.6.

7. P I ⊂ P and P I ⊂ I follow from Lemma 3.8.

8. Example 2.2 gives us the model of M and P but of neither E nor I. Example 3.4 gives us the model of E and I but of neither M nor P.

9. The rest of the relations from Table follows from the symmetry of ∥ and the duality of ⊂ and ⊃. □

Lemma 3.10. For any variety V from any of the classes 𝒜, ℬ, 𝒟, 𝒵 we have V ⊂Q.

Proof. We already concluded that Q is the greatest variety in Q_{4}. As it does not belong to 𝒜, we have M ⊂ Q, P ⊂ Q, E ⊂ Q, I ⊂ Q. The rest of relations follow from the transitivity of ⊂. □

Collected together, Lemmas 3.1–3.10 prove:

Theorem 3.1. The relationships given in Figure 1 are valid in the strong sense.

We aim to prove the same result for Figure 2. For that, we need more examples.

Example 3.7. Let (S; ∘) be a quasigroup with the Cayley table for the operation ∘ given in Table 3. It is a model of (Ub0) (with e = 0) but not of (C) because elements 1 and 2 do not commute.

Similarly, since 0 ∘ (1 ∘ 2) ≠ (2 ∘ 1) ∘ 0, (Ub1) is not true either.

Example 3.8. Let (S; ∘) be a quasigroup with the Cayley table for the operation ∘ given in Table 4. It is a model of (U) but not of (b0).

\begin{matrix}∘ & 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 0 & 3 & 4 & 2 \\ 2 & 2 & 4 & 0 & 1 & 3 \\ 3 & 3 & 2 & 4 & 0 & 1 \\ 4 & 4 & 3 & 1 & 2 & 0 \end{matrix}  \begin{matrix}∘ & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ 1 & 2 & 0 & 1 \\ 2 & 1 & 2 & 0 \end{matrix}

Table 3. A model of (Ub0) Table 4. A model of (U) but not of (C) but not of (b0)

Example 3.9. Let a multiplicative group S_{3} be given and let us define an operation / by x/y = xy^{−1} . Then the quasigroup (S; /) is a model of (Ub1) (with e = 0), but not of (B11) because (1/0)/(0/5) ≠ (5/0)/(0/1).

Lemma 3.11. The relationship between varieties from 𝒰 is given by the following table:

CU UB11 Ub1 Ub0 U
CU =  ⊂  ⊂  ⊂ ⊂
UB11 ⊃  =  ⊂  ∥ ⊂
Ub1 ⊃  ⊃  =  ∥ ⊂
Ub0 ⊃  ∥  ∥  = ⊂
U  ⊃  ⊃  ⊃  ⊃ =

Proof. 1. We have CU ⊆ UB11. If we force the operation · to be a T– quasigroup in (CU), (UB11), we get (BT 1), (D1) respectively. Since BT 1 and D1 are different, the same must be true for CU and UB11. Therefore, CU ⊂ UB11.

Also, CU ⊆ Ub0. Using model (S; ∘) from Example 3.7 we prove CU ⊂ Ub0.

2. UB11 ⊆ Ub1. By Example 3.9, UB11 ≠ Ub1.

Take a quasigroup from UB11 ∪ Ub0. If we apply unipotency in (B11) (with y = x), we get e · uv = vu · e and (using (b0)) e · uv = e · vu. Commutativity follows. Therefore, such quasigroup belongs to CU which is different from both UB11 and Ub0 proving UB11∥Ub0.

3. Taking Ub1 instead of UB11, we prove Ub1∥Ub0.

4. According to Example 3.8 Ub0 ⊂ U.

5. The rest of the relations are either trivial or follow by the transitivity of ⊂, or else by duality of ⊂ and ⊃. □

Lemma 3.12. The relationship between varieties from 𝒰 and 𝒬 is given by the following table:

C U1B11 B11 U1 Q
CU ⊂  ⊂  ⊂ ⊂ ⊂
UB11 ∥  ∥  ⊂  ∥ ⊂
Ub1 ∥  ∥  ∥ ∥ ⊂
Ub0 ∥  ∥  ∥ ∥ ⊂
U ∥  ∥  ∥ ∥ ⊂

Proof. 1. We have CU ⊆ U1B11. Since CU and U1B11 belong to 𝒰 and 𝒬 respectively they must be different, so CU ⊂ U1B11.

2. It is easy to see that the meet of UB11 with any of C, U1B11, U1 is CU which is different from any of them and consequently UB11∥C, UB11∥U1B11, UB11∥U1.

3. The meet of Ub1 and any of C, U1B11, U1 is CU. Therefore, Ub1∥C, Ub1∥U1B11 and Ub1∥U1.

The meet of Ub1 and B11 is UB11 which is different from both, so Ub1∥B11.

4. The meet of Ub0 and any of C, U1B11, B11, U1 is CU and consequently Ub0∥C, Ub0∥U1B11, Ub0∥B11, Ub0∥U1.

5. The case of U is analogous to 4.

6. The rest of the relations are trivial. □

Lemma 3.13. The relationship between varieties from 𝒬 is given by the following table:

C U1B11 B11 U1 Q
C  =  ∥  ⊂  ∥ ⊂
U1B11 ∥  =  ⊂ ⊂ ⊂
B11 ⊃  ⊃  =  ∥ ⊂
U1  ∥  ⊃  ∥ = ⊂
Q  ⊃  ⊃  ⊃ ⊃ =

Proof. Take the class 𝒬 = {Q, U1, U1B11, B11, C} and add assumption that all operations from all varieties are T-quasigroups. We get five varieties of quasigroups: the variety T of T-quasigroups (which is not a member of Q_{4}), D11, BT 11, T 11 and T 1. Moreover, this mapping is an order isomorphism. The relationships between elements of 𝒬 are determined by the relationships of their images in 𝒜 ∪ ℬ ∪ {Q} (replacing T by Q). □

Therefore we have:

Theorem 3.2. The relationships given in Figure 2 are valid in the strong sense.

The following Lemmas reveal relationships between varieties from 𝒵, 𝒟, ℬ, 𝒜 on one side and varieties from 𝒰, 𝒬 on the other.

Lemma 3.14. The relationship between varieties from 𝒟 and 𝒰 is given by the following table:

CU UB11 Ub1 Ub0 U ; D1 ∥  ⊂  ⊂  ∥ ⊂

Proof. A D1-quasigroup is of the form xy = Ax − Ay + e. Applying this to CU and Ub0 we get BT 1 which is different from all the three, proving D1∥CU and D1∥Ub0.

According to Lemma 1.1 D1 ⊆ UB11. Since D1 and UB11 belong to different classes, they are different. Therefore D1 ⊂ UB11. According to Lemma 3.13 and the transitivity of ⊂, we have D1 ⊂ Ub1 and D1 ⊂ U. □

Lemma 3.15. The relationship between varieties from 𝒟 and 𝒬 is given by the following table:

\begin{aligned}& C & U1B11 & B11 & U1 & Q \\ D1 & ∥ & ∥ & ⊂ & ∥ & ⊂ \end{aligned}

Proof. The meet of D1 and any of C, U1B11, U1 is BT 1. Therefore, D1 is incomparable to any of C, U1B11, U1.

Trivially, D1 ⊂ UB11 ⊂ B11 ⊂ Q. □

Lemma 3.16. For a V ∈ ℬ, W ∈ 𝒰 we have V ∥W.

Proof. Follows from ℬ∥𝒰. □

Lemma 3.17. The relationship between varieties from ℬ and 𝒬 is given by the following table:

C U1B11 B11 U1 Q
BT11 ∥  ⊂  ⊂ ⊂ ⊂
BM ∥  ∥  ∥ ∥ ⊂
BP ∥  ∥  ∥ ∥ ⊂

Proof. 1. The meet of BT 11 and C is BT 1 which is different from both and so BT 11∥C.

According to Lemma 1.2 BT 11 ⊆ U1B11 but, as they belong to different classes, they must be different. Consequently, BT 11 ⊂ U1B11. From transitivity, BT 11 ⊂ B11, BT 11 ⊂ U1 and BT 11 ⊂ Q.

2. The meet of BM and C is BT 1. Therefore, BM∥C.

The meet of BM and any of U1, B11, U1B11 is BT 11. Consequently BM is incomparable to any of them.

The relation BM ⊂ Q is trivially true.

3. The case with BP instead of BM is analogous. □

Lemma 3.18. For a V ∈ 𝒜, W ∈ 𝒰 we haveV ∥W.

Proof. Follows from 𝒜∥𝒰. □

Lemma 3.19. The relationship between varieties from 𝒜 and 𝒬 is given by the following table:

C U1B11 B11 U1 Q
T1 ⊂  ∥  ⊂  ∥ ⊂
T11 ∥  ∥  ⊂  ∥ ⊂
D11 ∥  ∥  ∥ ⊂ ⊂
ME ∥  ∥  ∥ ∥ ⊂
PI ∥  ∥  ∥ ∥ ⊂
M ∥  ∥  ∥ ∥ ⊂
P ∥  ∥  ∥ ∥ ⊂
E ∥  ∥  ∥ ∥ ⊂
I ∥  ∥  ∥ ∥ ⊂

Proof. 1. All T 1-quasigroups are commutative since A = B. From transitivity we have T 1 ⊂ B11 and T 1 ⊂ Q.

T 1 ∩ U1 = BT 1 and consequently T 1∥U1. T 1 ∩ U1B11 ⊆ T 1 ∩ U1 = BT 1. Therefore, T 1∥U1B11 as well.

2. According to Lemma 1.2 T 11 ⊆ B11. Since they belong to different classes 𝒜 and 𝒬 respectively, they must be different.

As T 11 ∩ C = T 1 we have T 11∥C.

T 11∩U1B11 = T 11∩U1 = BT 11 and consequently T 11∥U1B11 and T 11∥U1.

3. According to Lemma 1.2 D11 ⊆ U1. Since they belong to different classes 𝒜 and 𝒬 respectively, they must be different.

D11 ∩ C = BT 1 and D11 ∩ U1B11 = D11 ∩ B11 = BT 11, so D11∥C, D11∥U1B11, D11∥B11.

4. ME ∩ C = BT 1 and ME ∩ U1B11 = ME ∩ B11 = ME ∩ U1 = BT 11; therefore ME∥C, ME∥U1B11, ME∥B11, ME∥U1.

5. P I ∩ C = BT 1 and P I ∩ U1B11 = P I ∩ B11 = P I ∩ U1 = BT 11; therefore P I∥C, P I∥U1B11, P I∥B11, P I∥U1.

6. The meets of M and C, U1B11, B11, U1 are T 1, BT 11, T 11 and BT 11 respectively. This proves incomparability of M to any of C, U1B11, B11, U1.

7. Incomparability of E and I to C, U1B11, B11, U1 is proven similarly. □

Finally, we have to determine the relationship of LLU and RLU to each other and to all other varieties from Q_{4}.

Lemma 3.20. The varieties LLU and RLU are incomparable.

Proof. For a quasigroup · from LLU, x · y = Ax − Ay + c for appropriate A, + and c. If we apply this to an identity which determines RLU, for example xx · yz = uy · uz, we get commutativity and consequently D1. As D1 ∈ 𝒟 and is therefore different from both LLU, RLU, this implies LLU∥RLU. □

Lemma 3.21. We have D1 ⊂ LLU, D1 ⊂ RLU.

Proof. We have D1 ⊆ LLU and D1 ⊆ RLU. Since D1 belongs to 𝒟, while LLU, RLU belong to 𝒢, we infer D1 ⊂ LLU and D1 ⊂ RLU. □

Lemma 3.22. For a V ∈ ℬ, W ∈ 𝒢 we have V ∥W.

Proof. Follows from ℬ∥𝒢. □

Lemma 3.23. For a V ∈ 𝒜, W ∈ 𝒢 we have V ∥W.

Proof. Follows from 𝒜∥𝒢. □

Lemma 3.24. The relationship between varieties from 𝒢 and 𝒰 is given by the following table:

CU UB11 Ub1 Ub0 U
LLU ∥  ∥  ⊂  ∥ ⊂
RLU ∥  ∥  ⊂  ∥ ⊂

Proof. We have LLU ⊆ Ub1 ⊂ U. Since LLU belongs to 𝒢 and Ub1 belongs to 𝒰, it follows that LLU ⊂ Ub1 and LLU ⊂ U.

The meet of LLU and any of CU, UB11, Ub0 is BT 1. Therefore, LLU∥CU, LLU∥UB11, LLU∥Ub0.

The relationships for RLU follow from the left-right duality for groupoids. □

Lemma 3.25. The relationship between varieties 𝒢 and 𝒬 is given by the following table:

C U1B11 B11 U1 Q
LLU ∥  ∥  ∥ ∥ ⊂
RLU ∥  ∥  ∥ ∥ ⊂

Proof. The meet of LLU and any of C, U1B11, U1 is BT 1. Consequently, LLU∥C, LLU∥U1B11, LLU∥U1.

From LLU ∩ B11 = D1 it follows that LLU∥B11. The relationships for RLU follow from the left–right duality for groupoids. □

Using the symmetry of ∥ and duality of ⊂ and ⊃, we can complete the proof of the main theorem of the paper.

Theorem 3.3. The relationships given in Figure 4 are valid in the strong sense.

4. Conclusions

In [2], we explicitly promised to prove in this paper:

(1) That 19 varieties: Q, C, B11, U1, U, Ub0, Ub1, CU, LLU, RLU, M, P, E, I, ME, P I, T 1, D1, BT 1 are mutually distinct.

(2) That each of the seven varieties U1B11, UB11, T 11, D11, BM, BP and BT 11 (also mutually distinct, as well as different from above 19 varieties) can be axiomatized by two level identities with four variables (in the variety of quasigroups), cannot be axiomatized by a single level identity with four variables but can be axiomatized by the single level identity with eight variables.

(3) That the conjunction of any subset of 105 identities gives one of the above 26 varieties.

(4) That the ordering ’being a subset’ on the set Q_{4} of the above varieties is a lattice ordering. However, this lattice is not a sublattice of the lattice of all varieties of quasigroups.

Two more promisees were given elsewhere in [2]:

(5) That the proof of the independence of (U) and (U1) will be given.

(6) That the diagram of the lattice Q_{4} will be given. We can fulfill these promisses now.

Proof. (1) The proof is spread throughout Section 3.

(2) The seven varieties are defined in Table 1 by two level identities with four variables. The equivalence of these systems to some level identities with eight variables is hinted in the text on page 30. In Table 5 we give the correspondence of these varieties and some of the identities which define them.

variety  defining identity
U1B11 (xy · yx)(pq · rs) = (uv · vu)(sr · qp)
UB11 (xx · yy)(pq · rs) = (uu · vv)(sr · qp)
T 11 (xy · uv)(pq · rs) = (xu · yv)(sq · rp)
D11 (xy · ux)(pq · qr) = (vy · uv)(ps · sr)
BM (xy · uv)(pq · qr) = (xu · yv)(ps · sr)
BP (xy · uv)(pq · rp) = (vy · ux)(sq · rs)
BT 11 (xy · uv)(pq · qr) = (xu · yv)(rs · sp)

Table 5. Varieties of quasigroups–one identity with eight variables

As none of these systems is equivalent to above 19 identities, the varieties cannot be axiomatized by a single level identity with four variables. The proof that each of the seven varieties is different from any other in Q_{4} is also spread throughout Section 3.

(3) Follows from the induction and the closeness of Q_{4} under the meet operation.

(4) The lattice property can be verified in Figure 4 directly. The join of M and P in Q_{4} is Q. In the lattice of all varieties of quasigroups, the join of M and P must be a subvariety of the variety T of all T -quasigroups (as both M and P are T –quasigroups), but the variety Q is not a T –quasigroup.

(5) On account of Lemma 3.12 U∥U1. Independence follows.

(6) On account of Theorem 3.3, the lattice Q_{4} is given in Figure 4. □

5. Problems

The following problems suggest themselves:

Problem 1. Solve (systems of) quasigroup level equations with eight variables. Give the lattice Q_{8} of varieties determined by the corresponding identities.

Problem 2. Solve (systems of) quasigroup level equations with 2^{n} variables for a given n. Describe the lattice Q_{2^{n}} of varieties determined by the corresponding identities.

Problem 3. Solve (systems of) quasigroup level equations of any length. Describe the lattice Q_{∞} of varieties determined by corresponding identities.

The methods of this and other papers from the reference list of [2] are sufficiently strong to solve these problems. The real problem lays in finding the method to handle the combinatorial explosion borne by the growth of n. For example, the number of quadratic level equations with eight variables is 2 027 025.

We can always classify varieties in Q_{2^{n}} (Q_{∞} ) as we did in Section 2. There is a possibility that there is a new class of varieties with all quasigroups being group isotopes, but such that every variety contains a non–unipotent quasigroup. Let us call this ∼-class ℋ.

Problem 4. Is there a (nonempty) ℋ in Q_{2^{n}} (Q_{∞} )? If there is, what is the minimal n such that ℋ ≠ ∅?

References

1. W. Förg-Rob, A. Krapež, Equations which preserve the height of variables, Aequationes Math. 70 (2005), 63–76; DOI 10.1007/s00010-005-2790-x.

2. A. Krapež, Quadratic level quasigroup equations with four variables I, Publ. Inst. Math., Nouv. Sér 81(95) (2007), 53–67; DOI 102298/PIM0795053K

Mathematical Institute SASA (Received 08 05 2012) Kneza Mihaila 36 11001 Belgrade Serbia

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