Monadic second-order definability in weak arithmetics

I shall give a survey of monadic second-order definability in relatively weak arithmetical structures on $\mathbb{N}$, such as
$$ \langle \mathbb{N}; \leqslant \rangle , \quad \langle \mathbb{N}; +, = \rangle , \quad \langle \mathbb{N}; \,| \,\rangle , \quad \langle \mathbb{N}; \bot \rangle \quad \text{and} \quad \langle \mathbb{N}; \times, = \rangle $$
where $|$ and $\bot$ denote the divisibility relation and the coprimeness relation respectively. Moreover, if time permits, I shall mention some related results on first-order definability. The topic of this talk may be described as ‘weak arithmetics’, broadly understood.