Equality of left and right nil element
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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Suppose is a magma (a set with a binary operation ). Suppose is a left nil element (i.e., for all ) and is a right nil element for (i.e., for all ). Then, .
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.
Given: A magma with binary operation , a left nil element for , a right nil element for .
Proof: Consider the product . Since is a left nil element, . Since is a right nil element, . Thus, .