On the size of congruence lattices for models of theories with definability of congruences (Q793011)

The author studies the size of congruence lattices ConA of algebras A which are models of a given first order theory T. The author defines the following cardinals: \(C_ T(\lambda)=\sup \{| Con(A)|:A\vDash T,| A| =\lambda \}, c_ T(\lambda)=\inf \{| Con(A)|:A\vDash T,| A| =\lambda \}, L_ T(\lambda)=\sup \{length\quad of\quad Con(A):A\vDash T,| A| =\lambda \}, l_ T(\lambda)=\inf \{length\quad of\quad Con(A):A\vDash T,| A| =\lambda \}.\) The paper is organised as follows. Section 1. Introduction. Section 2. Notation and basic concepts. In Section 3 the author proves that for a theory with infinite models exactly one of the following cases may happen: Case 1. For every infinite cardinal \(\lambda\), \(L_ T(\lambda)=ded(\lambda)\leq C_ T(\lambda)\leq 2;\) Case 2. There exists a positive integer n such that, for every infinite cardinal \(\lambda\), \(L_ T(\lambda)=n\), \(C_ T(\lambda)=\lambda\); Case 3. There are positive integers m, n such that, for every infinite cardinal \(\lambda\), \(C_ T(\lambda)=m\), \(L_ T(\lambda)=n\). In Section 4, the author studies theories with the definability of compact congruences and the special case of \(\aleph_ 0\)-categorical theories. Section 5. Examples and counterexamples.