Submitted to Quasigroups and Related Systems

A class of quasigroups associated with a cubic Pisot number

Vedran Krčadinac and Vladimir Volenec

October 5, 2005

Abstract

In this paper idempotent medial quasigroups satisfying the identity (ab · a)a = b are studied. An example are the complex numbers with multiplication defined by a · b = (1 − q)a + qb, where q is a solution of q^{3} − 2q^{2} + q − 1 = 0. The positive root of this cubic equation can be viewed as a generalization of the golden ratio. It turns out that the quasigroups under consideration have many similar properties to the so-called golden section quasigroups.

1 Introduction

Let q ≠ 0, 1 be a complex number and define a binary operation on ℂ by a·b = (1−q)a+qb. It is known that (ℂ, ·) is an IM-quasigroup, i.e. satisfies the laws of idempotency and mediality:

a · a = a,  (1)
ab · cd = ac · bd.  (2)

Immediate consequences are the identities known as elasticity, left and right distributivity:

ab · a = a · ba,  (3)
a · bc = ab · ac,  (4)
ab · c = ac · bc.  (5)

2000 Mathematics Subject Classification: 20N05.

Keywords: IM-quasigroup, Pisot number.

This quasigroup will be denoted by C(q). For some special values of q, the quasigroup satisfies additional identities. If q = \frac{1+\sqrt{5}}{2} is the golden ratio, C(q) is a representative example of the golden section or GS-quasigroups. GS-quasigroups were defined in [8] as idempotent quasigroups satisfying the (equivalent) identities a(ab · c) · c = b, a · (a · bc)c = b; see also [2], [3], [4] and [10]. An alternative definition would be as IM-quasigroups with the simpler identity a(ab · b) = b. In this paper we study IM-quasigroups satisfying a similar identity:

(ab · a)a = b.  (6)

Representative examples are the quasigroups C(q) with q a root of q^{3} −

2q^{2}+q−1 = 0. Denote by r_{1,2} = \sqrt[3]{\frac{25±\sqrt{69}}{2}}. The roots of this cubic equation are q_{1} = \frac{1}{3} (2 + r_{1} + r_{2}) ≈ 1.755 and q_{2,3} = \frac{1}{6} (4 − r_{1} − r_{2} ± i\sqrt{3} (r_{1} − r_{2})) ≈ 0.123 ± 0.745 i. The number q_{1} is a Pisot number, i.e. an algebraic integer greater than 1 whose algebraic conjugates q_{2,3} have absolute values less than 1. This number was considered in [5] as a generalization of the golden ratio and was called the second upper golden ratio. Therefore, we will refer to IM-quasigroups satisfying the identity (6) as G_{2}-quasigroups.

In the context of [5], the second lower golden ratio was the positive root of p^{3} − p − 1 = 0. This is the smallest Pisot number p_{1} ≈ 1.325; note that q_{1} = p_{1}^{2}. For more details about Pisot numbers see [1].

In this paper it is shown that G_{2}-quasigroups have many properties similar to those of GS-quasigroups. For example, they allow a simple definition of parallelograms using an explicit formula for the fourth vertex. In the last section G_{2}-quasigroups are characterized in terms of Abelian groups with a certain type of automorphism.

2 Basic properties and further identities

The following lemma will be used quite often.

Lemma 2.1. In an IM-quasigroup, identity (6) is equivalent with either of the identities

(a · ba)a = b,  (7)
a(ba · a) = b.  (8)

Proof. By using elasticity we get (ab · a)a =^{(3)} (a · ba)a =^{(3)} a(ba · a).

G_{2}-quasigroups  3

Note that the equivalence holds even in a groupoid satisfying (1) and (2). Elasticity follows directly from idempotency and mediality, without using solvability or cancellativity. Consequently, the definition of G_{2}-quasigroups can be relaxed to the identities alone.

Proposition 2.2. Any groupoid satisfying (1), (2) and (6) is necessarily a quasigroup.

Proof. Given a and b define x = ab · a and y = ba · a. From (6) and (8) we see that xa = b and ay = b, i.e. the groupoid is left and right solvable. Now

assume ax_{1} = ax_{2} and y_{1}a = y_{2}a. Then, x_{1} =^{(6)} (ax_{1} · a)a = (ax_{2} · a)a =^{(6)} x_{2}

and y_{1} =^{(8)} a(y_{1}a · a) = a(y_{2}a · a) =^{(8)} y_{2}, so the groupoid is left and right cancellative.

The next proposition is similar to [8, Theorem 5].

Proposition 2.3. In a G_{2}-quasigroup, any two of the equalities ab = c, ca = d and da = b imply the third.

Proof. Denote the equalities by (i), (ii) and (iii), respectively. Then we have:

(i), (ii) ⇒ (iii) : da =^{(ii)} ca · a =^{(i)} (ab · a)a =^{(6)} b,
(i), (iii) ⇒ (ii) : ca =^{(i)} ab · a =^{(iii)} (a · da)a =^{(7)} d,
(ii), (iii) ⇒ (i) : ab =^{(iii)} a · da =^{(ii)} a(ca · a) =^{(8)} c.

We list some more identities valid in G_{2}-quasigroups. They are accompanied by pictures illustrating the example of the complex plane with multiplication defined by a · b = (1 − q_{1})a + q_{1}b.

Proposition 2.4. The following identity holds in any G_{2}-quasigroup:

(a · ab)c · a = ac · b.  (9)

Proof. (a · ab)c · a =^{(5)} (a · ab)a · ca =^{(5)} (a · ab)(ca) · (a · ca) =^{(3)} (a · ab)(ca) ·

(ac · a) =^{(2)} (ac)(ab · a) · (ac · a) =^{(4)} ac · (ab · a)a =^{(6)} ac · b.

Figure 1: Identity (9) in the complex plane.

Proposition 2.5. The following identity holds in any G_{2}-quasigroup:

(ab · a)c · b = (ab · c)a.  (10)

Proof. (ab·a)c·b =^{(5)} (ab·b)(ab)·cb =^{(3)} (ab)(b·ab)·cb =^{(2)} (ab·c)·(b·ab)b =^{(7)} (ab·

c)a.

Figure 2: Identity (10) in the complex plane.

Proposition 2.6. The following identity holds in any G_{2}-quasigroup:

a · (ba · c)d = b(ac · d).  (11)

Proof. a · (ba · c)d =^{(5)} a · (ba · d)(cd) =^{(4)} (a · ba)(ad) · (a · cd) =^{(2)} (a · ba)a · (ad ·

cd) =^{(7)} b(ad · cd) =^{(5)} b(ac · d).

Figure 3: Identity (11) in the complex plane.

3 Parallelograms and other geometric concepts

The points a, b, c, d of a medial quasigroup are said to form a parallelogram, denoted by Par(a, b, c, d), if there are points p, q such that pa = qb and pd = qc. In [7] it was proved that this relation satisfies the axioms of parallelogram space:

1. For any three points a, b, c there is a unique point d such that Par(a, b, c, d).

2. Par(a, b, c, d) implies Par(e, f, g, h), where (e, f, g, h) is any cyclic permutation of (a, b, c, d) or (d, c, b, a).

3. Par(a, b, c, d) and Par(c, d, e, f) imply Par(a, b, f, e).

In an IM-quasigroup, the unique point d of axiom 1 satisfies the following equation [9, Theorem 12]:

ab · dc = ac.  (12)

This equation can be explicitly solved for d in GS-quasigroups: d = a·b(ca·a) [8, Theorem 6]. Here we prove a similar result for G_{2}-quasigroups.

Proposition 3.1. In a G_{2}-quasigroup, for any a, b, c we have

Par(a, b, c, (ba · cb)b).

Proof. By substituting d = (ba · cb)b into the equation (12) we get

ab · [(ba · cb)b · c] = ac.

It suffices to show that this is a valid identity in any G_{2}-quasigroup:

ab · [(ba · cb)b · c] =^{(5)} ab · [(ba · c)(cb · c) · bc] =^{(2)} ab · [(ba · c)b · (cb · c)c] =

=^{(6)} ab · [(ba · c)b · b] =^{(5)} ab · [(ba · b) · cb]b =^{(5)} ab · [(ba · b)b · (cb · b)] =

=^{(6)} ab · a(cb · b) =^{(4)} a · b(cb · b) =^{(8)} ac.

Now we have a direct definition of parallelograms in G_{2}-quasigroups, without using auxiliary points:

Par(a, b, c, d) ⟺ d = (ba · cb)b.  (13)

Using the parallelogram relation geometric concepts such as midpoints, vectors and translations can be introduced. Of course, in the special case of the quasigroups C(q) the concepts agree with the usual definitions of plane geometry. Thus, geometric theorems can be proved by formal calculations in a quasigroup. We give an example particular to G_{2}-quasigroups (Theorem 3.4).

In any medial quasigroup, b is said to be the midpoint of the pair of points a, c if Par(a, b, c, b) holds. This is denoted by M(a, b, c). The following proposition provides a characterization in G_{2}-quasigroups.

Proposition 3.2. In a G_{2}-quasigroup, M(a, b, c) is equivalent with

c = (ab · ba)a.  (14)

Proof. By axiom 2 of parallelogram spaces, M(a, b, c) is equivalent with Par(b, a, b, c), and the claim follows from (13).

To facilitate notation, we introduce a new binary operation:

a ∗ b = (ba · a)b.  (15)

Starting from the quasigroup C(q_{1}), this defines the binary operation in the quasigroup C(p_{1}), i.e. a ∗ b = (1 − p_{1})a + p_{1}b. If ab = c (resp. a ∗ b = c), we say that b divides the pair of points a, c in the second upper (resp. lower) golden ratio. Here are some properties of the new binary operation. It is assumed that the original binary operation has higher priority than ‘∗’, e.g. a ∗ bc means a ∗ (bc).

Figure 4: A new binary operation defined by (15).

Lemma 3.3. The operation defined by (15) in a G_{2}-quasigroup satisfies the following identities:

a ∗ a = a,  (16)
ab ∗ cd = (a ∗ c)(b ∗ d),  (17)
(a ∗ (a ∗ b)c)c = b.  (18)

Proof. Idempotency of the new operation (16) follows directly from (1). Identity (17) follows by repeated application of mediality:

\begin{aligned}ab ∗ cd & =^{(15)} & (cd · ab)(ab) · cd =^{(2)} (ca · db)(ab) · cd =^{(2)} (ca · a)(db · b) · cd = \\ & =^{(2)} & (ca · a)c · (db · b)d =^{(15)} (a ∗ c)(b ∗ d). \end{aligned}

Here is the proof of identity (18):

(a ∗ (a ∗ b)c)c =^{(15)} {[(ba · a)b · c]a · a}[(ba · a)b · c] · c =
=^{(2)} {[(ba · a)b · c]a · (ba · a)b}(ac) · c =
=^{(2)} {[(ba · a)b · c](ba · a) · ab}(ac) · c =
=^{(2)} {[(ba · a)b · ba](ca) · ab}(ac) · c =
=^{(5)} {[(ba · b)(ab) · ba](ca) · ab}(ac) · c =
=^{(2)} {[(ba · b)b · (ab · a)](ca) · ab}(ac) · c =
=^{(6)} {[a(ab · a) · ca](ab) · ac}c =^{(2)} {[ac · (ab · a)a](ab) · ac}c =
=^{(6)} [(ac · b)(ab) · ac]c =^{(5)} [(ac · a)b · ac]c =^{(2)} [(ac · a)a · bc]c =
=^{(6)} (c · bc)c =^{(7)} b.

Identity (17) could be called mutual mediality of the two binary operations. By identifying two factors various kinds of distributivities follow: a ∗ bc = (a ∗ b)(a ∗ c), a(b ∗ c) = ab ∗ ac and their right counterparts. Identity (18) is an analogue of the defining identity for GS-quasigroups [8]. It is used in the proof of the following theorem.

Theorem 3.4. In a G_{2}-quasigroup, suppose that a ∗ e = c, a ∗ f = b and cg = f. Then, bg = e. Furthermore, suppose M(a, h, g) and h ∗ g = d. Then, dh = a and M(b, d, c).

Proof. The first claim follows by substitution:

bg = (a ∗ f)g = (a ∗ cg)g = (a ∗ (a ∗ e)g)g =^{(18)} e.

If, in addition, M(a, h, g) and h ∗ g = d hold, we get g = (ah · ha)a by (14), and the remaining claims follow by tedious, but straightforward computations:

dh = (h ∗ g)h = [h ∗ (ah · ha)a]h =^{(15)} {[(ah · ha)a · h]h · (ah · ha)a}h =
=^{(2)} {[(ah · ha)a · h](ah · ha) · ha}h =^{(2)} {[(ah · ha)a · ah](h · ha) · ha}h =
=^{(5)} {[(ah · a)(ha · a) · ah](h · ha) · ha}h =
=^{(2)} {[(ah · a)a · (ha · a)h](h · ha) · ha}h =
=^{(6)} {[h · (ha · a)h](h · ha) · ha}h =^{(4)} {h[(ha · a)h · ha] · ha}h =
=^{(5)} {h[(ha · h)(ah) · ha] · ha}h =^{(2)} {h[(ha · h)h · (ah · a)] · ha}h =
=^{(6)} [h · a(ah · a)](ha) · h =^{(4)} h[a(ah · a) · a] · h =^{(3)} h[a · (ah · a)a] · h =
=^{(6)} (h · ah)h =^{(7)} a.

To prove M(b, d, c), we utilize (14) once more:

(bd · db)b =^{(4)} (bd · d)(bd · b) · b =^{(5)} (bd · d)b · (bd · b)b =^{(6)} (bd · d)b · d =
= (bd · d)b · (h ∗ g) =^{(15)} (bd · d)b · (gh · h)g =
=^{(2)} (bd · d)(gh · h) · bg =^{(2)} (bd · gh)(dh) · bg =
=^{(2)} (bg · dh)(dh) · bg = (ea · a)e =^{(15)} a ∗ e = c.

In the special case of the quasigroup C(q_{1}), Theorem 3.4 proves:

Corollary 3.5. Let ABC be a triangle and suppose the points E and F divide \overline{AC} and \overline{AB} in the second lower golden ratio, respectively. Then the cevians \overline{BE} and \overline{CF} intersect in a point G that divides them in the second upper golden ratio. Furthermore, the midpoint H of \overline{AG} divides the third cevian \overline{AD} in the second upper golden ratio.

Figure 5: Geometric interpretation of Theorem 3.4.

The statement of Corollary 3.5 remains true if every instance of the second lower/upper golden ratio is replaced by the corresponding n-th golden ratio (for a definition see [5]). For n = 1, both the lower and the upper golden ratio are equal to \frac{1+\sqrt{5}}{2} and we get the geometric interpretation of [8, Theorem 15].

4 Representation theorems

Let (G, +) be an Abelian group with an automorphism φ such that the following equality holds for every x ∈ G:

φ^{3}(x) − 2φ^{2}(x) + φ(x) − x = 0.  (19)

Define another binary operation on G by the formula

a · b = a + φ(b − a).  (20)

It is easy to verify that G is an IM-quasigroup with this new operation. Furthermore, the identity (6) follows from (19):

(ab · a)a = ab · a + φ(a) − φ(ab · a)
= ab + φ(a) − φ(ab) + φ(a) − φ(ab) − φ^{2}(a) + φ^{2}(ab)
= 2φ(a) − φ^{2}(a) + ab − 2φ(ab) + φ^{2}(ab)
= 2φ(a) − φ^{2}(a) + (id − 2φ + φ^{2})(a + φ(b) − φ(a))
= [a − φ(a) + 2φ^{2}(a) − φ^{3}(a)] + [φ^{3}(b) − 2φ^{2}(b) + φ(b) − b] + b
=^{(19)} b.

Therefore, (G, ·) is a G_{2}-quasigroup. The purpose of this section is to show that any G_{2}-quasigroup can be obtained in this way.

Theorem 4.1. Let (G, ·) be a G_{2}-quasigroup. Choose an arbitrary o ∈ G and define a new binary operation on G by the formula

a + b = (oa · bo)o.  (21)

Then, (G, +) is an Abelian group with neutral element o.

Proof. We first prove associativity, commutativity and that o is the neutral element:

(a + b) + c =^{(21)} [o · (oa · bo)o](co) · o =^{(5)} [o · (oa · bo)o]o · (co · o) =
=^{(7)} (oa · bo)(co · o) =^{(2)} (ob · ao)(co · o) =^{(2)} (ob · co)(ao · o) =
=^{(7)} [o · (ob · co)o]o · (ao · o) =^{(5)} [o · (ob · co)o](ao) · o =
=^{(2)} (oa)[(ob · co)o · o] · o =^{(21)} a + (b + c),

a + b =^{(21)} (oa · bo)o =^{(2)} (ob · ao)o =^{(21)} b + a,

a + o =^{(21)} (oa · oo)o =^{(1)} (oa · o)o =^{(6)} a.

For any a ∈ G define −a = o · (o · oa)a. This is the inverse of a:

\begin{aligned}a + (−a) & =^{(21)} & \{oa · [o · (o · oa)a]o\}o =^{(5)} (oa · o)\{[o · (o · oa)a]o · o\} = \\ & =^{(6)} & (oa · o) · (o · oa)a =^{(2)} (oa)(o · oa) · oa =^{(7)} o. \end{aligned}

Theorem 4.2. The mappings φ : x ↦ ox and ψ : x ↦ xo are automorphisms of the group (G, +) of Theorem 4.1 and satisfy the identity

ψ(a) + φ(b) = ab.  (22)

Proof. The following shows that φ is an automorphism:

\begin{aligned}φ(a) + φ(b) & = & oa + ob =^{(21)} (o · oa)(ob · o) · o =^{(3)} (o · oa)(o · bo) · o = \\ & =^{(4)} & o(oa · bo) · o =^{(3)} o · (oa · bo)o =^{(21)} o(a + b) = φ(a + b). \end{aligned}

The proof that ψ is an automorphism is similar. Finally,

\begin{aligned}ψ(a) + φ(b) & = & ao + ob =^{(21)} (o · ao)(ob · o) · o =^{(3)} (o · ao)(o · bo) · o = \\ & =^{(4)} & o(ao · bo) · o =^{(5)} o(ab · o) · o =^{(7)} ab. \end{aligned}

Theorem 4.3. Equations (19) and (20) are satisfied in the setting of the previous two theorems.

Proof. As a special case of (22), we see that ψ(x) + φ(x) = xx =^{(1)} x, i.e. ψ(x) = x − φ(x). Now equation (20) follows directly from (22):

ab = ψ(a) + φ(b) = a − φ(a) + φ(b) = a + φ(b − a).

To prove equation (19), note that

ψ^{2}(x) = ψ(x − φ(x)) = x − φ(x) − φ(x − φ(x)) = φ^{2}(x) − 2φ(x) + x.

Therefore, φ^{3}(x) − 2φ^{2}(x) + φ(x) = φ(ψ^{2}(x)) = o(xo · o) =^{(8)} x.

This is a direct proof of a G_{2}-version of Toyoda’s representation theorem for medial quasigroups [6].

References

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[6] K. Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (1941), 221–227.

[7] V. Volenec: Geometry of medial quasigroups, Rad Jugoslav. Akad. Znan. Umjet. 421 (1986), 79–91.

[8] V. Volenec: GS-quasigroups, Časopis Pěst. Mat. 115 (1990), no. 3, 307–318.

[9] V. Volenec: Geometry of IM-quasigroups, Rad Jugoslav. Akad. Znan. Umjet. 456 (1991), 139–146.

[10] V. Volenec and Z. Kolar-Begović : Affine regular pentagons in GS-quasigroups, Quasigroups Related Systems 12 (2004), 103–112.

Department of Mathematics University of Zagreb Bijenička 30 HR-10000 Zagreb Croatia

e-mail: krcko@math.hr, volenec@math.hr