Equality of left and right nil element

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This article gives a statement (possibly with proof) of how, if a left-based construction and a right-based construction both exist, they must be equal.
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Statement

Suppose (S,*) is a magma (a set S with a binary operation *). Suppose n_1 is a left nil element (i.e., n_1 * a = n_1 for all a \in S) and n_2 is a right nil element for S (i.e., a * n_2 = n_2 for all a \in S). Then, n_1 = n_2.

Proof

Proof idea

A left nil element can be thought of as an element that dominates the product when placed on the left, and a right nil element is an element that dominates the product when placed on the right. To show that these are equal, we need to pit the left and right nil elements against each other. Since both of them must dominate, they must both be equal.

Formal proof

Given: A magma S with binary operation *, a left nil element n_1 for *, a right nil element n_2 for *.

To prove: n_1 = n_2

Proof: Consider the product n_1 * n_2. Since n_1 is a left nil element, n_1 * n_2 = n_1. Since n_2 is a right nil element, n_1 * n_2 = n_2. Thus, n_1 = n_2.