trait Semigroup {
    fn compose(&self, other: &Self) -> Self;
}

trait Monoid: Semigroup {
    fn identity() -> Self;
}

Any semigroup can trivially be made into a monoid by adding an identity element:

impl<T: Semigroup + Clone> Semigroup for Option<T> {
    fn compose(&self, other: &Self) -> Self {
        match self {
            Some(ref a) => match other {
                Some(ref b) => Some(a.compose(b)),
                None => self.clone()
            },
            None => other.clone()
        }
    }
}

impl<T: Semigroup + Clone> Monoid for Option<T> {
    fn identity() -> Self {
        None
    }
}

Now we have our interface for an eager arqtree.

pub struct ArqTree<T> {
    val: Vec<T>
}

impl<T: Monoid + Clone> ArqTree<T> {
    fn modify(&mut self, pos: usize, f: fn(&T) -> T) {
        // implement modify
    }
    
    fn query(&self, l: usize, r: usize) -> T {
        // implement query
    }
}

However, this will not suffice when we need to do range updates!