Theories admitting congruences over sets and boundedness

Samuel Braunfeld, Michael C Laskowski

Published 2021-09-18Version 1

We consider several ways of decomposing models into parts of bounded size forming a congruence over a base, and show that admitting any such decomposition is equivalent to mutual algebraicity at the level of theories. We also show that a theory $T$ is mutually algebraic if and only if for every $M \preceq N \models T$, there is an absolute bound on the number of types realized in $N-M$ over $M$.