Let $A$ be an algebra with a bilinear map $m: A \times A \to A$ (multiplication). So we define the center of $A$ as the commutative elements of $A$:
$$Z(A) = \{ a \in A \mid a.a' = a' . a, ~ \forall a' \in A \}$$ (The idea can be extended to another algebraic structures, like groups and rings.)
I want to know if there is a terminology/symbol/definition of the elements of $A$ that are associatives, or flexibles, or alternatives. Ex:
$$F(A) = \{ a \in A \mid a.(b.a) = (a.b).a, ~ \forall b \in A \}$$