A sufficient condition for nonpresentability of structures in hereditarily finite superstructures

We introduce a class of existentially Steinitz structures containing, in particular, the fields of real and complex numbers. A general result is proved which implies that if $\mathfrak M$ is an existentially Steinitz structure then the following structures cannot be embedded in any structure $\Sigma$-presentable with trivial equivalence over $\mathbb{HF}(\mathfrak M)$: the Boolean algebra of all subsets of $\omega$, its factor modulo the ideal consisting of finite sets, the group of all permutations on $\omega$, its factor modulo the subgroup of all finitary permutations, the semigroup of all mappings from $\omega$ to $\omega$, the lattice of all open sets of real numbers, the lattice of all closed sets of real numbers, the group of all permutations of $\mathbb R$ $\Sigma$-definable with parameters over $\mathbb{HF(R)}$, and the semigroup of such mappings from $\mathbb R$ to $\mathbb R$.