Which of the following sets are monoids under addition?

Question

Which of the following are monoids under addition?

i) All integers

ii) All even integers

iii) All odd integers

iv) all positive even integers

v) all integers or halves of integers

Refer to answer below for attempt.

Answer 1

i) The integers are an Abelian Group under addition, and hence are most certainly a monoid under addition.

ii) The even integers are associative and have the identity $0$

iii) The odd integers do not hold under closure, since there exist $a,b,c$ with $a+b=c$, $a,b\in\Bbb Z_o$ but $c\not\in\Bbb Z_o$.

iv) The positive even integers don't have an identity element. Assume they do, $$2+x=x+2=2$$ $$x+2=2\implies x=2-2=0,0\not\in\Bbb Z^+_e$$ Hence there is no element satisfying the identity relation for $2\in\Bbb Z^+_e$ and hence there is no identity element, e.g. this is not a monoid under addition.

v) All integers and half integers are a monoid under addition since we have an isomorphic mapping $f:\Bbb Z^* \to \Bbb Z$, $f(x)=2x$ which takes this integers and half integers to $\Bbb Z$. Hence this is an Abelian group under addition.

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Thus only iii and iv are not monoids under addition.