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.?

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    $\begingroup$ Yes and no. As in any structure, you have the notion of substructure generated by a single element. In associative structures, this behaves well. Actually, say in a group, you have two kinds of substructures: subgroups, and normal subgroups (= $G$-subgroups, $G$ being viewed endowed with $G$-action). In the latter case, there is no particular good behavior of single-generated substructures. In semigroups, you in addition have ideals, and single-generated behave well. In magmas, single-generated substructures or left/two-sided ideals don't behave well. Not simpler than 2-generated ones, say. $\endgroup$ – YCor May 10 at 21:56
  • $\begingroup$ @YCor The purpose of the question is mostly methodological. It may be helpful to understand a principle on the simplest structure, and then to find that it behaves well in a more complex structure. $\endgroup$ – Alex C May 10 at 22:16
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    $\begingroup$ I guess that the notion of principal ideal is not likely to be very helpful in magmas or non-associative algebras in general, and doesn't look simpler than the notion of ideal generated by 2 elements. $\endgroup$ – YCor May 10 at 22:21
  • $\begingroup$ @YCor A definition of a non-associative ideal generated by a set of elements would be even better. Are there any references on the item? $\endgroup$ – Alex C May 10 at 22:39
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    $\begingroup$ I expect you'd get something such as an inductive definition (as you'd give to a computer), and nothing simpler. Do it inside a magma $M$ is enough (then extend by linearity): the 2-sided ideal generated by $Y\subset M$ is the union over all $n$ of all possible $n$-fold products involving at least one element of $Y$. Recursively, define $M_1=M$, $Y_1=Y$, $M_n=\bigcup_{p+q=n}M_pM_q$, $Y_n=\bigcup_{p+q=n}(M_pY_q\cup Y_pM_q)$ with $p,q>0$ in unions. Then the ideal generated by $Y$ is $\bigcup_{n\ge 1} Y_n$. $\endgroup$ – YCor May 10 at 22:51

In a magma $M$, one can describe the 2-sided ideal generated by a subset $Y$ as follows: define by induction $$M_1=M,\;Y_1=Y,\; M_n=\bigcup_{p,q\ge 1,p+q=n}M_pM_q,\;Y_n=\bigcup_{p,q\ge 1,p+q=n}(M_pY_q\cup Y_pM_q).$$ Then the 2-sided ideal generated by $Y$ is $Y_\infty=\bigcup_{n\ge 1} Y_n$.

An alternative definition, is to define $Y'_1=1$, $Y'_{n+1}=Y'_nM\cup MY'_n\cup Y'_nY'_n$; then $\bigcup_{n\ge 1} Y'_n=Y_\infty$.

If $R$ is a scalar ring (=commutative associative unital) and $A$ is an $R$-algebra (not assumed associative), if $Y$ is a subset of $A$, one can define $Y_\infty$ as previously (using only multiplication). Then the $R$-submodule generated by $Y_\infty$ equals the 2-sided ideal generated by $Y$. In case $A$ is unital, this is also the additive subgroup generated by $Y_\infty$.

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