Left nucleus

Definition

For a magma $(S,*)$, the left nucleus is the set of $a\in S$ such that:

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

Elements of the left nucleus are termed left-associative elements or left nuclear elements.

The left nucleus of a magma is a submagma. In fact, it is a subsemigroup that contains any left neutral element of the magma and also any left nil of the magma. For full proof, refer: left-associative elements of magma form submagma