In An Introduction to Category Theory, exercise 1.3.6(b), Simmons writes,
Consider any pair of Pos-arrows... Show that if arrow f is inverse of arrow g, then, f(g(f)) = f && g(f(g)) = g and hence g(f) is a closure operation on A and f(g) is a co-closure operation on B.
The solution my study group got was,
Idendity of S =< f(g) && g(f) =< Identity of T f =< f(g(f)) && g(f(g)) =< g therefore f = f(g(f))
which looks like an inverse, but not quite. Is this what Simmons means by closure? Any suggestions welcome.
(I apologize that I don't know how to write tex; editors please clean up my presentation. Thanks to all in advance for the clarifications.)