The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity.
However, the definitions of a principal ideal I know (for a semigroup or a ring) assume associativity.
For example, the left principal ideal $S^1a$ of a semigroup $S$ is an ideal because $S^1(S^1a) = (S^1S^1)a$.
The two-sided principal ideal of a semigroup is the set $S^1aS^1$ which is defined due to associativity. https://en.wikipedia.org/wiki/Green%27s_relations
I am trying to generalize the definition of a principal ideal to non-associative structures.
Would it be a correct generalization to say that a principal ideal is an ideal that can be obtained by taking a single element and all finite products with the element as one of the oprands?
The left (right) principal ideal of a non-associative magma $M$ generated by an element $a$ is the set that includes $a$ and all finite products of elements of $M$ where $a$ is the rightmost (resp. leftmost) operand.
The two-sided principal ideal of $M$ generated by $a$ is the set that includes $a$ and all finite products of elements of $M$ that contain $a$ as an operand.
I am wondering if there is a notation for the sets of finite products for a non-associative structure.
Instead of $S^1a$ or $S^1aS^1$ it must include all possible combinations of $...S^1S^1a$ or $...S^1S^1aS^1S^1...$.
Are there better approaches or formulations of a principal ideal of a non-associative magma?
Are there generalizations of a principal ideal for non-associative rings, algebras, etc.?