A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M, \star)$ such that $M$ is a set and $\star$ is a partial binary operation on $M$ (I say that $\mathbb M$ is a magma if $\star$ is total). But what about partial semigroups? At least in principle, many alternative definitions are possible: The only thing I would take for certain is that a partial semigroup must be a partial magma $\mathbb M = (M, \star)$ for which $\star$ satisfies some kind of associativity, and of course I've my personal list. Specifically, I say that $\mathbb M$ is

1. (properly) associative if for all $x,y,z \in M$ such that $(x \star y) \star z$ and $x \star (y \star z)$ are defined, it holds $(x \star y) \star z = x \star (y \star z)$.
2. left pre-associative if for all $x,y,z \in M$ such that $x \star y$ and $y \star z$ is defined, it holds that "$(x \star y) \star z$ is defined" implies "$x \star (y \star z)$ is defined and $(x \star y) \star z = x \star (y \star z)$".
3. right pre-associative if the dual of $\mathbb M$ is left pre-associative.
4. pre-associative if it is both left and right pre-associative.
5. strongly associative if for all $x,y,z \in M$ it holds that "$x \star y$ and $y \star z$ are defined" implies "$(x \star y) \star z$ and $x \star (y \star z)$ are defined, and also $(x \star y) \star z = x \star (y \star z)$".

In this taxonomy (which doesn't aim to be complete by any means), "being (propertly) associative" corresponds to the weakest possible form of associativity, in the sense that it is implied by all the others. Moreover, all of the above properties collapse into each other if $\mathbb M$ is a magma. So, the (somewhat philosophical) question is:

What should it be a partial semigroup? Do you envisage any "higher logic" advocating for one instead of another choice?

My own answer is that a partial semigroup should be a strongly associative partial magma, in the sense of the above condition 5. But, on the one hand this doesn't seem to be the "standard" definition in the literature (see, e.g., R.H. Schelp, A partial semigroup approach to partially ordered sets, Proc. London Math. Soc. (1972), s3-24 (1), 46-58, where partial semigroups are pre-associative partial magmas, in the sense of the above condition 4), and on the other hand I can't give myself a reason why this should be better or worse than something different (which bothers me much...).

A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M, \star)$ such that $M$ is a set and $\star$ is a partial binary operation on $M$ (I say that $\mathbb M$ is a magma if $\star$ is total). But what about partial semigroups? At least in principle, many alternative definitions are possible: The only thing I would take for certain is that a partial semigroup must be a partial magma $\mathbb M = (M, \star)$ for which $\star$ satisfies some kind of associativity, and of course I've my personal list. Specifically, I say that $\mathbb M$ is

1. (properly) associative if for all $x,y,z \in M$ such that $(x \star y) \star z$ and $x \star (y \star z)$ are defined, it holds $(x \star y) \star z = x \star (y \star z)$.
2. left pre-associative if for all $x,y,z \in M$ such that $x \star y$ and $y \star z$ is defined, it holds that "$(x \star y) \star z$ is defined" implies "$x \star (y \star z)$ is defined and $(x \star y) \star z = x \star (y \star z)$".
3. right pre-associative if the dual of $\mathbb M$ is left pre-associative.
4. pre-associative if it is both left and right pre-associative.
5. strongly associative if for all $x,y,z \in M$ it holds that "$x \star y$ and $y \star z$ are defined" implies "$(x \star y) \star z$ and $x \star (y \star z)$ are defined, and also $(x \star y) \star z = x \star (y \star z)$".
6. left dissociative if for all $x,y,z \in M$ it holds that "$(x \star y) \star z$ is defined" implies "$x \star (y \star z)$ is defined and $(x \star y) \star z = x \star (y \star z)$".
7. right dissociative if the dual of $\mathbb M$ is left dissociative.
8. dissociative if it is both left and right dissociative.

In this taxonomy (which doesn't aim to be complete by any means), "being (propertly) associative" corresponds to the weakest possible form of associativity, in the sense that it is implied by all the others. Moreover, all of the above properties collapse into each other if $\mathbb M$ is a magma. So, the (somewhat philosophical) question is:

What should a partial semigroup be? Do you envisage any "higher logic" advocating for one instead of another choice?

My own answer is that a partial semigroup should be a strongly associative partial magma, in the sense of the above condition 5. But, on the one hand this doesn't seem to be the "standard" definition in the literature (see, e.g., R.H. Schelp, A partial semigroup approach to partially ordered sets, Proc. London Math. Soc. (1972), s3-24 (1), 46-58, where partial semigroups are pre-associative partial magmas, in the sense of the above condition 4), and on the other hand I can't give myself a reason why this should be better or worse than something different (which bothers me much...).