Hello every body , I have a question in categories. Please help me
to solve it.

Let f:X --> Y be a M-mapping then show that :
1) f is injective <--> f is monic,
2) f is surjective <--> f is epic.

Two definitions are needed:

1)Let M be a semigroup , and X be a nonempty set ,X is called a M-
set if a function like g:XxM-->X exists such that for each m and n
in M and x in X ,

(i) g(x,mn)=g(g(x,m),n)
(ii) g(x,1)=x

2)If X and Y are two M-sets , then the function f:X--> Y is called a
M-mapping provided that for each m in M and x in X ,

f(x,m)=f(x).m

Best regards ,
Sara Raja.