Let
be a set and
the set of all binary
operations on
.
We say that
is a distributive set of operations if all pairs of elements
are right
distributive, that is,
(we allow
).
The question of which groups can be realized as distributive sets
was asked by J. Przytycki. The initial guess that embedding into
for some
holds for any
was complicated by
an observation that if
is idempotent (), then
commutes with every
element of
. The first
noncommutative subgroup of
(the group
)
was found in October 2011 by Y. Berman.
Here we show that any group can be embedded in
for
(as a
set). We also discuss minimality of embeddings observing, in particular, that
with six elements is the
smallest set such that
contains a nonabelian subgroup.
Keywords
monoid of binary operations, distributive set, shelf,
multishelf, distributive homology, embedding, group
