category of monoids

Properties from the database

Deduced properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

—

isomorphisms: bijective homomorphisms
monomorphisms: injective homomorphisms
epimorphisms: A monoid map $f : T \to S$ is an epimorphism iff $S$ equals the dominion of $U := f(T) \subseteq S$ , meaning that for every $s \in S$ there are $u_1,\dotsc,u_{m+1} \in U$ , $v_1,\dotsc,v_m \in U$ , $x_1,\dotsc,x_m \in S$ and $y_1,\dotsc,y_m \in S$ such that $s = x_1 u_1$ , $u_1 = v_1 y_1$ , $x_{i-1} v_{i-1} = x_i u_i$ , $u_i y_{i-1} = v_i y_i$ , $x_m v_m = u_{m+1}$ and $u_{m+1} y_m = s$ .
regular monomorphisms:
regular epimorphisms: surjective homomorphisms