Title: Partial monoids
Abstract: A partial monoid is a set A with a multiplication and a unit, but the multiplication is only defined on a certain subset of pairs. This means the multiplication m:AxA -> A is a partial map, a map that is only defined on a certain subset of its domain. The category of finite commutative partial monoids is an important ingredient for the construction of tangent infinity categories where a bicategory of spans of partial monoids encodes a weakly commutative ring structure. As partial maps are generalized by restriction categories one can directly generalize the definition of a partial monoid in any restriction category.
I will present a certain restriction category B of bags (a.k.a. multisets) with a partial monoid in it. Any partial monoid in any join restriction category X is induced through a functor from B to X which means the partial monoid in B is the generic partial monoid. This characterization of partial monoids is analogous to the concept of a Lawvere theory of an algebraic structure.