Unbiased partial monoids

We introduce unbiased partial monoids as special case of unbiased lax monoids.

In an unbiased partial monoid, for each natural number n, the partial n-ary multiplication map can have its own domain. For example, we can encode semigroups as the subclass of unbiased partial monoids where the 0-ary multiplication map is everywhere undefined and all other multiplication maps are total.

We look at the definition of unbiased partial monoids from several perspectives and consider what actions of such monoids should look like.