TITLE: Partial $\mathsf{T}$-algebras, where $\mathsf{T}$ is a Lawvere theory
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: The closest reference to what you are asking for that I know is

Erik Palmgren and Steve Vickers: Partial Horn logic and cartesian categories. Annals of Pure and Applied Logic. Volume 145, Issue 3, March 2007, Pages 314-353. (DOI, PDF)

To get the universal partial algebra going one has to address partiallity in a serious way. For instance, it matters how complicated the domains of definition may be, or to put it another way, what is the language for expressing definedness conditions? The above paper addresses this, which is why it the title talks about "Horn logic" but is really (also) about universal partial algebra.
Another place to look is essentially algebraic theories.