Amalgamation property

Examples include in modal logic as an incestual accessibility relation,[clarification needed] and in lambda calculus as a manner of reduction having the Church–Rosser property.

Recall that f: A → B is an embedding if f is an injective morphism which induces an isomorphism from A to the substructure f(A) of B.

In general, the amalgamation property can be considered for a category with a specified choice of the class of morphisms (in place of embeddings).

The counterexample for this starts with L1 containing a single element e and extends in two different ways to L3, one in which e is the smallest and the other in which e is the largest.

Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of e. Now consider the class of algebraically closed fields.