# Principal ideal of a non-associative magma

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.?

• 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. – YCor May 10 at 21:56
• @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. – Alex C May 10 at 22:16
• 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. – YCor May 10 at 22:21
• @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? – Alex C May 10 at 22:39
• 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 $\bigcup_{n\ge 1} Y_n$. – 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 $$Y_\infty$$.