Free medial quandles
cka, Agata Pilitowska, and Anna Zamojska-Dzienio
Abstract. This paper gives the construction of free medial quandles as well as free
n-symmetric medial quandles and free m-reductive medial quandles.
A binary algebra (Q, ·) is called a rack if the following conditions hold, for
every x, y, z ∈ Q:
• x(yz)=(xy)(xz) (we say Q is left distributive),
• the equation xu = y has a unique solution u ∈ Q (we say Q is a left
An idempotent rack is called a quandle (we say Q is idempotent if xx = x for
every x ∈ Q). A quandle Q is medial if, for every x, y, u, v ∈ Q,
An important example of a medial quandle is an abelian group A with an
operation ∗ deﬁned by a ∗ b = (1 − h)(a)+h(b), where h is an automorphism
of A. This construction is called an aﬃne quandle (or sometimes an Alexander
quandle) and denoted by Aﬀ(A, h). In the literature [6, 7], the group A is
usually considered to be a Z[t, t
]-module, where t · a = h(a), for each a ∈ A.
We shall adopt this point of view here as well and we usually write Aﬀ(A, r)
instead, where r is a ring element.
Note that in universal algebra terminology, an algebra is said to be aﬃne if
it is polynomially equivalent to a module. A subreduct of an aﬃne algebra is
called a quasi-aﬃne algebra, see e.g., . Clearly, aﬃne quandles are quasi-
Medial quandles lie in the intersection of the class of quandles and the class
of modes . Recent development in quandle theory is motivated by knot
theory (see e.g., [1, 4]). The knot quandle is a very powerful knot invariant.
Quandles also have applications in diﬀerential geometry  and graph the-
ory . Modes are generally idempotent and entropic algebras (algebras with
a commutative clone of term operations). Mediality is another name for en-
tropicity in the binary case. For a more detailed history of medial quandles,
we refer to .
Presented by P. Dehornoy.
Received April 14, 2016; accepted in ﬁnal form July 15, 2016.
2010 Mathematics Subject Classiﬁcation: Primary: 08B20; Secondary: 15A78, 20N02.
Key words and phrases: quandles, medial quandles, binary modes, free algebras.
Algebra Univers. 78 (2017) 43–54
Published online May 23, 2017
© 2017 The Author(s)
This article is an open access publication