3
$\begingroup$

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?

$\endgroup$
4

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.