A Sufficient Condition for Nonpresentability of Structures in Hereditarily Finite Superstructures

Abstract

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 Σ-presentable with trivial equivalence over ℍ\( \mathbb{F} \)(\( \mathfrak{M} \)): the Boolean algebra of all subsets of ω, its factor modulo the ideal consisting of finite sets, the group of all permutations on ω, its factor modulo the subgroup of all finitary permutations, the semigroup of all mappings from ω to ω, the lattice of all open sets of real numbers, the lattice of all closed sets of real numbers, the group of all permutations of ℝ Σ-definable with parameters over ℍ\( \mathbb{F} \)(ℝ), and the semigroup of such mappings from ℝ to ℝ.