A class of -structures has the amalgamation property if and only if whenever and are elementary embeddings for then there is some and some elementary embeddings for so that for all . That is, the following diagram commutes.
Compare this with the free product with amalgamated subgroup for groups and the definition of pushout contained there.
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & {A} \ar[dl]_{f_1} \ar[dr]^{f_2} & \ {B_1} \ar[dr]_{g_1} & & {B_2} \ar[dl]^{g_2} \ & {C} & } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/3964/l2h/img11.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & {A} \ar[dl]_{f_1} \ar[dr]^{f_2} & \ {B_1} \ar[dr]_{g_1} & & {B_2} \ar[dl]^{g_2} \ & {C} & } } \end{xy}$")