Representation and preservation of the metamathematical quotients of A. Robinson (Q1802263)

The concept of metamathematical ideal in structures, developed by A. Robinson, generalizes the notion of (algebraic) congruence in two ways: it takes into account the predicates of the language, and also it is defined relatively to a given theory. A concept of quotient follows. This paper solves the problem of characterizing the theories preserving these quotients, raised by \textit{A. F. Margarit} and \textit{L. M. Laita} [Z. Math. Logik Grundlagen Math. 33, 497-505 (1987; Zbl 0639.03036)]. It is found that such theories are those equivalent to a union of Positive and of Universal Strict Horn sentences (by showing first that the class \(Q(T^*)\) of the quotients of all models of \(T\) is \(H(T^*)\cap\text{SP}(T^*)\)).