Categories can be presented in the language of arrows only, without reference to objects (as discussed, say, here: Categories presented with Arrows only, no objects: partial monoids), and this is a partial monoid. Next question appears naturally - can such a partial monoid be completed up to a total monoid?
My first impulse is to add a new arrow "$\oslash$" ("null"), and set $g \circ f = \oslash$, for any two arrows $f, g$, such that $dom(g) \ne codom(f)$, then extend the composition operation in such a manner that $h \circ \oslash = \oslash$ and $\oslash \circ h = \oslash$, for any $h$. Also, I would discurd all identities and use instead only one arrow $1$, such that $h \circ$ $1 = 1$ and $1$ $\circ$ $h = h$, for any $h$. Am I missing something, or this is the required completion to a monoid?
If this is a completion to a monoid, and it contains an element which behaves like a "zero" in a ring, then next question appears naturally: cannot this monoid be extended to a ring?
Has anybody encountered in literatue something similar to this?