Hope you are well.
I am studying flow networks (flows of water) in graph theory. By flow network I mean a graph $G = (V,E)$ where $V$ is a set of $ℝ_{>0}$ labelled vertices (water tanks filled with $ℝ_{>0}$ liters of water) and $E$ is a set of $V$'s $ℝ_{>0}$ weighted edges (flows of $ℝ_{>0}$ liters of water), a subset of $V × V$, together with a positive partial function $f: V × V → ℝ_{>0}$ called the water flow function that sends $ℝ_{>0}$ liters of water, over time, from one vertex (water tank) to another subject to water availability in the predecessor vertex (water tank).
How would you name this semigroupoid-like structure in abstract algebra, with closure ($*$ is a partial operation and therefore not defined for every $a$ and $b$ in SG due to some water tanks not able to send water without receiving water first) and identity (a water tank sending water to itself) not required, that is:
- Always "right invertible" and not always "left invertible". What do I mean by this? for all $a$ in SG (semigroupoid), $a * a′$ is always defined but $a′ * a$ is not (a water tank, right after, can always receive what it originally sent, but may not have enough water to send first and then receive).
- Not always associative.
- Not always commutative.
The elements of this semigroupoid-like structure would be the members of the cartesian product of the underlying set and itself less the identity members. For example, the elements of this semigroupoid-like structure defined over a set of $3$ members (water tanks) $[a, b, c]$ would be the $3*3-3$ ordered pairs $(a, b)$, $(a, c)$, $(b, a)$, $(b, c)$, $(c, a)$, and $(c, b)$ (water flow partial functions). Therefore, an element will "act" by sending water from one water tank to another one.
The operation $*$ would be that of partial function composition. That is, applying one ordered pair after another (one flow after another). Again, this operation would not be defined for every pair of ordered pairs.
Flow network: https://en.wikipedia.org/wiki/Flow_network
Semigroupoid: https://en.wikipedia.org/wiki/Semigroupoid
Thanks a lot!