On $\Sigma$-definability of hereditarily finite and list superstructures

This work is concerned with properties of hereditarily finite superstructures $\mathbb{HF}(\mathfrak{M})$ and hereditarily finite list superstructures $\mathbb{HW}(\mathfrak{M})$. The main result states that any relation $\Sigma$-definable in a hereditarily finite superstructure $\mathbb{HF}(\mathfrak{M})$ can also be defined by $\Sigma$-formula in a hereditarily finite list superstructure $\mathbb{HW}(\mathfrak{M})$ and vice versa.