Affiliation:
1. Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology , 50-370 Wroclaw , Poland
2. Flat 10, Albert Mansions, Crouch Hill , London N8 9RE , United Kingdom
Abstract
Abstract
Any pentagonal quasigroup
Q
Q
is proved to have the product
x
y
=
φ
(
x
)
+
y
−
φ
(
y
)
xy=\varphi \left(x)+y-\varphi (y)
, where
(
Q
,
+
)
\left(Q,+)
is an Abelian group,
φ
\varphi
is its regular automorphism satisfying
φ
4
−
φ
3
+
φ
2
−
φ
+
ε
=
0
{\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0
and
ε
\varepsilon
is the identity mapping. All Abelian groups of order
n
<
100
n\lt 100
inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity
(
x
y
⋅
x
)
y
⋅
x
=
y
\left(xy\cdot x)y\cdot x=y
is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is
{
1
1
n
:
n
=
0
,
1
,
2
,
…
}
\left\{1{1}^{n}:n=0,1,2,\ldots \right\}
. We prove that the only translatable commutative pentagonal quasigroup is
x
y
=
(
6
x
+
6
y
)
(
mod
11
)
xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11)
. The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group
Z
n
{{\mathbb{Z}}}_{n}
and its automorphism
φ
(
x
)
=
a
x
\varphi \left(x)=ax
is proved to determine the value of
a
a
and the range of values of
n
n
.