category of monoids

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

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Special morphisms

  • Isomorphisms: bijective homomorphisms
  • Monomorphisms: injective homomorphisms
  • Epimorphisms: A monoid map f:T→Sf : T \to S is an epimorphism iff SS equals the dominion of U:=f(T)⊆SU := f(T) \subseteq S, meaning that for every s∈Ss \in S there are u1,…,um+1∈Uu_1,\dotsc,u_{m+1} \in U, v1,…,vm∈Uv_1,\dotsc,v_m \in U, x1,…,xm∈Sx_1,\dotsc,x_m \in S and y1,…,ym∈Sy_1,\dotsc,y_m \in S such that s=x1u1s = x_1 u_1, u1=v1y1u_1 = v_1 y_1, xi−1vi−1=xiuix_{i-1} v_{i-1} = x_i u_i, uiyi−1=viyiu_i y_{i-1} = v_i y_i, xmvm=um+1x_m v_m = u_{m+1} and um+1ym=su_{m+1} y_m = s.