Let $\mathfrak{A}$ be an infinite algebraic structure without order relation for not $\omega$-categorical theory. For example, $\mathfrak{A}$ contains an arbitrary functional symbol $f^{(n)}$ and an arbitrary predicate symbol $P^{(m)}$. I am trying to define some fixed point operator (inflationary fixed point or partial fixed point) on $\mathfrak{A}$ such that the values at all steps are pairwise different ($X^i \ne X^j$ for all $i, j$). Is this possible? Are there any theorems about this?
Thank you.
Not always. For example, suppose all relations hold on all tuples and all the functions are just projection onto the first coordinate (and we allow arbitrarily many constant symbols). For every finite sublanguage $\Sigma$ of the language of $\mathfrak{A}$, the reduct $\mathfrak{A}\upharpoonright\Sigma$ is essentially trivial in the sense that there is a finite set $A_\Sigma\subseteq\mathfrak{A}$ so that every permutation of $\mathfrak{A}$ which is the identity on $A_\Sigma$ is an automorphism of $\mathfrak{A}\upharpoonright\Sigma$. Consequently there are no interesting inductive operators, even allowing finitely many parameters.