A NOTE ON MEDIAL QUASIGROUPS

VLADIMIR VOLENEC AND VEDRAN KRČADINAC

Abstract. In this short note we prove two results about medial quasigroups. First, let φ and ψ be binary operations defined by multiplication, left and right division in a medial quasigroup. Then φ and ψ are mutually medial, i.e. φ(ψ(a, b), ψ(c, d)) = ψ(φ(a, c), φ(b, d)). Second, four points a, b, c, d in an idempotent medial quasigroup form a parallelogram if and only if d = (a / b)(b \ c).

Let (Q, ·) be a quasigroup. We will denote left and right division in Q by \ and /, i.e. a \ c = b ⟺ ab = c ⟺ c / b = a. By a formula we mean any expression built up from a number of variables using the operations ·, \ and /. More precisely:

(1) the variables x, y, … are formulae;

(2) if φ, ψ are formulae, then so are φ · ψ, φ \ ψ and φ / ψ. A formula φ containing (at most) two variables gives rise to a new binary operation Q×Q → Q, which we will also denote by the letter φ.

A quasigroup is medial if the identity ab · cd = ac · bd holds. It is known that (Q, ·) is medial if and only if either (and therefore both) of the quasigroups (Q, \) and (Q, /) are medial. Our first result states that binary operations defined by formulae in a medial quasigroup are mutually medial:

Theorem 1. Let φ, ψ be the binary operations defined by any two formulae in a medial quasigroup Q. Then the following identity holds:

φ(ψ(a, b), ψ(c, d)) = ψ(φ(a, c), φ(b, d)).

Proof. It is easily verified that the operations ·, \ and / are mutually medial, i.e. the identities ab \ cd = (a \ c)(b \ d), ab / cd = (a / c)(b / d) and (a \ b) /(c \ d) = (a / c) \(b / d) hold. For example, if we denote x = a \ c and y = b \ d, then ax = c and by = d. Using mediality we get cd = ax · by = ab · xy ⟹ ab \ cd = xy = (a \ c)(b \ d).

Date: May 23, 2006.

2000 Mathematics Subject Classification. 20N05.

Key words and phrases. Medial quasigroup, parallelogram.

A binary operation defined by a formula φ is mutually medial with multiplication, left and right division:

φ(ab, cd) = φ(a, c) · φ(b, d),  (1)
φ(a \ b, c \ d) = φ(a, c) \ φ(b, d),  (2)
φ(a / b, c / d) = φ(a, c) / φ(b, d).  (3)

This is obvious for φ(a, b) = a and φ(a, b) = b, and follows by induction for more complicated formulae. Supposing the identities are true for φ_{1} and φ_{2}, we see that they also hold for φ = φ_{1} · φ_{2}:

\begin{aligned}φ(ab, cd) & = φ_{1}(ab, cd) · φ_{2}(ab, cd) = φ_{1}(a, c)φ_{1}(b, d) · φ_{2}(a, c)φ_{2}(b, d) = \\ & = φ_{1}(a, c)φ_{2}(a, c) · φ_{1}(b, d)φ_{2}(b, d) = φ(a, c) · φ(b, d). \end{aligned}

The argument is similar for identities (2), (3) and formulae φ = φ_{1} \ φ_{2}, φ = φ_{1} / φ_{2}.

Finally, mutual mediality of φ and ψ is obtained by induction on ψ:

φ(ψ(a, b), ψ(c, d)) = φ(ψ_{1}(a, b)ψ_{2}(a, b), ψ_{1}(c, d)ψ_{2}(c, d)) =^{(1)}
= φ(ψ_{1}(a, b), ψ_{1}(c, d)) · φ(ψ_{2}(a, b), ψ_{2}(c, d)) =
= ψ_{1}(φ(a, c), φ(b, d)) · ψ_{2}(φ(a, c), φ(b, d)) =
= ψ(φ(a, c), φ(b, d)).

Identity (2) is used if ψ = ψ_{1} \ ψ_{2}, and identity (3) if ψ = ψ_{1} / ψ_{2}. □

Corollary 2. If (Q, ·) is a medial quasigroup, then the binary operation defined by a formula φ is also medial.

Of course, (Q, φ) need not be a quasigroup. Special cases of Theorem 1 and Corollary 2 have been used in [1] and [7]. Identity (1) was proved earlier by Puharev [2].

In [3], some geometric concepts have been introduced in a medial quasigruop Q. For example, the points a, b, c, d ∈ Q are said to form a parallelogram, denoted by Par(a, b, c, d), if there are points p, q ∈ Q such that pa = qb and pd = qc. This quaternary relation satisfies the axioms of parallelogram space (for definitions and further references see [8]). In particular, given any three points a, b, c ∈ Q there is a unique d ∈ Q such that Par(a, b, c, d). In [5], the parallelogram relation in idempotent medial quasigroups (satisfying the additional identity aa = a) was characterized in several more direct ways. In even more special quasigroups, explicit formulae for the fourth vertex d of a parallelogram as a function of a, b and c are known; see [1], [4], [6] and [9]. Here we give such a formula valid in a general IM-quasigroup.

Theorem 3. Let Q be an idempotent medial quasigroup and a, b, c, d ∈ Q. Then, Par(a, b, c, d) holds if and only if there are x, y ∈ Q such that xb = a, by = c and xy = d.

Proof. Let x, y ∈ Q be elements satisfying xb = a, by = c and xy = d. By taking p = a and q = x, we see that pa = qb and pd = xb · xy = x · by = qc, i.e. Par(a, b, c, d) holds.

Now suppose Par(a, b, c, d) holds and denote x = a / b, y = b \ c. Then, xb = a and by = c. According to [3, Corollary 5], for any p ∈ Q there is a unique q ∈ Q such that pa = qb and pd = qc. Specially, for p = a we see that a = qb ⟹ q = x and ad = qc = xc = x · by = xb · xy = a · xy. Cancelling a from the left yields xy = d. □

Corollary 4. In an idempotent medial quasigroup, Par(a, b, c, d) holds if and only if d = (a / b)(b \ c).

Formulae for the fourth vertex of a parallelogram in hexagonal, quadratical, GS and G_{2}-quasigroups contain only multiplication. In fact, such formulae follow form Corollary 4 because left and right division can be expressed by multiplication in these four classes of quasigroups. The formulae given in [1], [4] and [9] are shorter because other identities valid in these particular classes of quasigroups were taken into account.

References

[1] V. Krčadinac, V. Volenec, A class of quasigroups associated with a cubic Pisot number, Quasigroups Related Systems 13(2005), 269–280.

[2] N.K. Puharev, Geometric questions of certain medial quasigroups (Russian), Sibirsk. Mat. Zh. 9(1968), 891–897.

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

[4] V. Volenec, GS-quasigroups, Časopis Pěst. Mat. 115(1990), 307–318.

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

[6] V. Volenec, Hexagonal quasigroups, Arch. Math. (Brno) 27a(1991), 113–122.

[7] V. Volenec, Quadratical groupoids, Note Mat. 13(1993), 107–115.

[8] V. Volenec, Parallelogram spaces and their corresponding algebraic and geometric structures, Math. Commun. 2(1997), 15–20.

[9] V. Volenec, R. Kolar-Šuper, Parallelograms in quadratical quasigroups, preprint.

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

  E-mail address: vladimir.volenec@math.hr
  E-mail address: vedran.krcadinac@math.hr