The paper gives a survey of results from the theory of medial and more general quasigroups and its connection with the geometry of point reflections.
The geometric theory of point reflections starts with two axiom systems \(\{S1-S4, S5\}\) or \(\{S1-S4, S6\}\) for the universal geometry \(\mathcal{V}\) of the Euclidean operations of point reflection and midpoint. These axioms are based on one sort of individual variables, to be interpreted as points, and two binary operation symbols \(\cdot \), with \(a \cdot b\) standing for “the reflection of \(b\) in \(a\)”, and \(\mu \), with \(\mu (a, b)\) standing for “the midpoint of segment \(ab\)”.
Quasigroups are also models of a certain axiom system in a language with the three binary operations \(\cdot\), \(\mu\) (both defined as above) and \(\nu \). In this context \(\mu(a,b)\) corresponds to the right division \(a/b\) and \(\nu(a,b)\) corresponds to the left division \(a\setminus b\). Moreover, the left cancellation law holds whenever \(S4\) holds, and the right cancellation law holds whenever \(S3\) holds.
The axioms \(S5\) and \(S6\) are equivalent over the theory defined by the axioms \(S1-S4\). Here, the axiom \(S5\) is the identity \(d\cdot (c\cdot (b\cdot a))=b\cdot (c\cdot (d\cdot a))\), and \(S6\) is the medial law \( (a\cdot b)\cdot (c\cdot d)=(a\cdot c)\cdot (b\cdot d)\). The authors give a new proof of this equivalence using purely syntactic derivations.
It is still an open question how many variables are needed for the axiomatization of \(\mathcal{V}\). Some aspects of the 3-variable theory and its connections with certain quasigroups are discussed. In particular, from the laws \(S5\) and \(S6\) one obtains 3-variable sentences by setting two variables equal, among them the left and the right distributive law for quasigroups. The authors also discuss the geometric meaning of these laws. A theorem of Kepka on certain anti-commutative quasigroups provides the following result: The geometry \(\mathcal{V}\) together with an additional property can be axiomatized by axioms containing at most 3 variables.
Next some representation theorems for certain classes of quasigroups are considered: first Toyoda’s representation theorem for medial quasigroups, then repesentation theorems for left Bruck loops and commutative Moufang loops. While Toyoda’s theorem provides a representation theorem for quasigrops satisfying the medial law \(S6\), here a representation theorem for quasigroups satisfying \(S5\) is given. It is shown that a quasigroup with \(S5\) is principally isotopic to an abelian group and is right linear on an abelian group. Moreover, an idempotent quasigroup with \(S5\) is a medial quasigroup.
In addition, quasigroups with the “reverse” property of \(S5\), namely \(((uz)y)x=((ux)y)z\), are characterized. It turns out that a quasigroup with this property is also principally isotopic to an abelian group and is left linear on an abelian group. An idempotent quasigroup with this property is medial. A quasigroup with both \(S5\) and its reverse is linear on an abelian group.
The last part of the paper is devoted to generalizations of \(S5\) by letting different binary operations take the place of the one \(\cdot \) operation occurring in \(S5\). These generalizations provide theorems for quasigroups without any known connection with geometry.