Left-associative elements of magma form submagma

Statement

Let $(S,*)$ be a magma (a set $S$ with binary operation $*$). Call an element $a\in S$ a left-associative element if the following holds:

$a*(b*c)=(a*b)*c\mathrm{\forall }b,c\in S$

Then:

1. The set of left-associative elements of $S$ forms a submagma of $S$.
2. If $S$ contains a left neutral element $e$, then $e$ is left-associative, and is a left neutral element for the submagma of left-associative elements.
3. If $S$ contains a left nil $n$, then $n$ is left-associative.

Related facts

• Middle-associative elements of magma form submagma
• Right-associative elements of magma form submagma

All these proofs make crucial use of the associativity pentagon: the pentagon describing the relation between the five different ways of associating a product of length four.

Proof

Proof of (1)

Given: A magma $(S,*)$, two left-associative elements $a_1,a_2\in S$

To prove: $a_1*a_2$ is left-associative

Proof: We need to show that, for any $b,c\in S$, we have:

$(a_1*a_2)*(b*c)=((a_1*a_2)*b)*c$

Let's do this. Start with the left side and proceed as follows:

$(a_1*a_2)*(b*c)=a_1*(a_2*(b*c))=a_1*((a_2*b)*c)=(a_1*(a_2*b))*c=((a_1*a_2)*b)*c$

Proof of (2)

Given: A magma $(S,*)$ with left neutral element $e$.

To prove: $e$ is in the submagma of left-associative elements and is left neutral for the submagma.

Proof: For any $b,c\in S$, $e*(b*c)=b*c=(e*b)*c$, so $e$ is left-associative. Since $e*b=b$ for all $b\in S$, $e*b=b$ for all left-associative elements in particular.

Proof of (3)

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