MAT246 mathematics
mathematics
MAT246: Homework Problems
Instructions: These problems should be completed individually. The usual rules for collaboration apply.
Your submission should be your own written work and should be typeset using LATEX (template here),
then submitted to Gradescope. Please provide full justification for all your answers.
Problem 1. Let (M, ·, e) be a monoid where M is finite.
(a) Show that if x ∈M , then there exists integers n ≥ 0 and k > 0 such that xn+k = xn.
Hint. Consider the set S = {xm ∈M | m ∈ N}. How many elements does S have?
(b) Show that for all x ∈M there exists an integer N > 0 such that x2N = xN .
Hint. Show that if n, k are as in (a), then xn+nk = xn.
(c) Using (b), conclude that M is a group if and only if the only element x ∈ M such that x2 = x is the
identity, e.
Problem 2.
(a) Prove that there exists a monoid M such that for every monoid N , there is a unique monoid homomor-
phism M → N .
(b) Prove that there exists a monoid M such that for every monoid N , there is a unique monoid homomor-
phism N →M .
(c) Show that the monoid you found in part (a) is “unique up to isomorphism of monoids”. That is, if M ′ is
a monoid satisfying the same property as M in (a), then M is isomorphic to M ′.
Note. This also holds for part (b)!