Notions of basis and span in a magma

linear algebramagma

Suppose that $C$ is a set with closure under the binary operation $+$. $(C,+)$ is therefore a magma.

I am trying to figure out if notions of basis, or span make sense in a magma.

Spanning set (?)

Suppose that I can find a subset of C: $C_s \subset C$, which has a finite number of elements: $C_s = \{c_{s_1}, \ldots, c_{s_n}\}$, so that every element of $C$ can be expressed as a "sum" of the elements of $C_s$.
Since my "sum" is not associative or commutative, I can have the following:

$c_i = ((c_{s_4} + c_{s_1}) + c_{s_4}) + c_{s_2}$

  1. Is $C_s$ necessarily closed under +?

  2. Since my $+$ is not associative or commutative, I could potentially have an infinite number of operands correct?

    2.1 subsidiary question: is there a clean/short way to write a non-associative non-commutative binary operation on a possibly infinite number of elements? Something like $\sum_k^n …$ but which clearly shows that there can be repeating elements (like $c_{s_4}$ above) and that the "sum" is not commutative/associative?

Basis

Let us now assume that this subset $C_s$ is made of "linearly independent" elements, so that I could ultimately define a basis in my magma.

  1. How would I properly define this "linear independence" property in my magma with a non-associative and non-commutative +?

I am familiar with the definition of a basis in a vector space E on K where:

$\forall x \in E, \exists \lambda_1, \ldots, \lambda_p \in K: x = \lambda_1e_1 + \ldots + \lambda_pe_p$

and

$\lambda_1e_1 + \ldots + \lambda_pe_p = 0 \implies \lambda_1 = 0, \ldots, \lambda_p = 0$

but I have problems connecting this with the magma

Best Answer

Suppose that I can find a subset of C: $C_s \subset C$, which has a finite number of elements: $C_s = \{c_{s_1}, \ldots, c_{s_n}\}$, so that every element of $C$ can be expressed as a "sum" of the elements of $C_s$.

Making "sum" precise here would be to say that every $c\in C$ can be obtained from the elements of $C_s$ by finitely many $+$ operations. You would then call $C_s$ a generating set of $(C,+)$

  1. Is $C_s$ necessarily closed under +?

No it is not, consider the magma $(\mathbb Z_{>0}, +)$ with generating set $\{1\}$.

  1. Since my $+$ is not associative or commutative, I could potentially have an infinite number of operands correct?

No, you can only have finitely many operands. Just like in any structure without a notion of limits, you only have finite products, finite sums, finite linear combinations, etc.

Of course, you can have arbitrarily many operands in a finite (non-associative) sum.

2.1 subsidiary question: is there a clean/short way to write a non-associative non-commutative binary operation on a possibly infinite number of elements? Something like $\sum_k^n ...$ but which clearly shows that there can be repeating elements (like $c_{s_4}$ above) and that the "sum" is not commutative/associative?

You can draw finite binary rooted trees with leaves labelled by elements of the magma. You have to encode the order of operations (the brackets) one way or another.

  1. How would I properly define this "linear independence" property in my magma with a non-associative and non-commutative +?

The closest you can get is saying a subset $I\subseteq C$ is independent if every $c\in I$ can not be generated from $I\setminus\{c\}$. This is an analogy with a subset of a vector space being linearly independent iff no vector can be written as a linear combination of the others.

You could then define a "basis" to be an independent generating set, but do not expect any nice properties from this.

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