Middle nucleus

Definition
For a magma $$(S,*)$$, the middle nucleus is the set of elements $$b \in S$$ such that:

$$a * (b * c) = (a * b) * c \ \forall \ a,c \in S$$

The elements of the middle nucleus are termed middle-associative elements or middle nuclear elements.

The middle nucleus of a magma is a submagma that is in fact a semigroup under the induced operation.

Related submagma-defining functions

 * Left nucleus
 * Right nucleus
 * Nucleus