Question. Has this proposal (or anything like it) for a category of partial $\mathsf{T}$-algebras been considered anywhere, and if so, where, and if not, does it have any obvious flaws or deficits?
Let $\mathsf{T}$ denote a Lawvere theory with generating object $\Box$.
I was thinking that a reasonable definition of "partial $\mathsf{T}$-algebra" might go something like this.
Firstly, let $\mathbf{PSet}$ denote the category of pointed sets. View $\mathbf{PSet}$ as a symmetric monoidal category with $\otimes$ as smash product, and note that there's reasonable-looking choices for projection maps like $X \otimes Y \rightarrow X$ and diagonals like $X \rightarrow X\otimes X$.
Secondly, define that a morphism of pointed sets $f : (X,\bot_X) \rightarrow (X,\bot_Y)$ is tight iff $f^{-1}(\bot_Y) = \{\bot_X\}$. See here for relevant nonsense.
Thirdly, make $\mathbf{PSet}$ into a (thin) $2$-category by defining that for each pair of morphisms $$f,g : (X,\bot_X) \rightarrow (Y,\bot_Y),$$ we have $f \leq g$ iff $$\forall x \in X(f(x) \neq \bot_Y \rightarrow f(x)=g(x)).$$
Fourthly, define that a partial $\mathsf{T}$-algebra is an oplax functor $F : \mathsf{T} \rightarrow \mathbf{PSet}$ with data $F(\Box^n) \cong (F\Box)^{\otimes n}$, and which turns each canonical projection $\pi_i : \Box^n \rightarrow \Box$ into the corresponding canonical projection $\pi_i:(F\Box)^{\otimes n} \rightarrow F\Box$ and the same for the diagonals.
And fifthly, define that a morphism $X \rightarrow Y$ of partial $\mathsf{T}$-algebras is just a lax natural transformation $f:X \rightarrow Y$ whose component at $\Box$ is tight. By "lax", mean that $$f_\Box \circ X(j) \leq Y(j) \circ f_{{\Box}^{\otimes n}}.$$
It would be nice, for example, if the partial $\mathsf{T}$-algebras for $\mathsf{T}$ equal to the Lawvere theory of commutative monoids were precisely the partial commutative monoids. This is probably too much to hope for, but at least its a guiding example.
