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.