A straight set is an equivariant set which is equivariantly isomorphic to a disjiont union of sets of the form . A straight semigroup is an equivariant semigroup whose universe is a straight set. We raise the following:
Question 1: is every equivariant, orbit-finite semigroup an image of some orbit-finite, straight semigroup under an equivariant mapping?
We show some simple preliminary observations towards the above question.
First, let us see why the answer to the question is not trivially true. Let be an equivariant semigroup. It is true that there is a straight set
, equivariant mappings
and
, such that
for all
. However, it is not clear that the mapping
can be chosen to be associative.
Example. We show an example which is not orbit-finite, but demonstrates some problems with the naive approach. Let be the set of all finite subsets of
, equipped with the operation
, where
is the symmetric difference of
and
.
is an equivariant semigroup, although not an orbit-finite one. There is the obvious surjection
which maps a tuple of distinct atoms to its support. The set
is straight (although not orbit-finite). The question is whether there is an associative operation
on it which lifts the operation
, so that
. A natural candidate attempt would be to define
as the tuple of those elements which appear an even number of times in the tuple
, ordered according to their order of appearance in
. However, this operation is not associative, as verified by the triple
,
,
. Similarly, one can show that there is no associative operation
on
such that
.
Example. Any equivariant semigroup (such as
above) can be straightened, by a non orbit-finite set, as follows. Choose any straight set
and equivariant surjective mapping
. Then
with concatenation is a straight semigroup, and the mapping
given by
is a surjective homomorphism from
to
.
Question 2. Does the following property hold. If is an image of a straight semigroup under an equivariant mapping, then it is an image of a straight semigroup under an equivariant surjection
which preserves least supports, i.e.
.
In particular, for the semigroup , is there an equivariant epimorphism which preserves least supports from a straight semigroup?