LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity

Statement

Let $*$ be a binary operation on a set $S$ (Binary operation on a set; associativity, commutativity, and a subset closed under the operation). If $e \in S$ is a left identity and $f \in S$ is a right identity for $*$ (Left identity, right identity, and two-sided identity for a binary operation), then $e = f$ . Consequently:

$*$ has at most one two-sided identity;
if $*$ has a two-sided identity $e$ , then $e$ is the only left identity and the only right identity for $*$ .

No associativity is assumed.

Facts & Assumptions

Given: A binary operation $*$ on a set $S$ , an element $e \in S$ with $e * x = x$ for all $x \in S$ , and an element $f \in S$ with $x * f = x$ for all $x \in S$ (Left identity, right identity, and two-sided identity for a binary operation).

[A1]

$e$ is a left identity: $e * x = x$ for every $x \in S$ .

[A2]

$f$ is a right identity: $x * f = x$ for every $x \in S$ .

Proof

technique · direct

1.1

Evaluate $e * f$ using [A1] with $x = f$ : $e * f = f$ .

1.2

Evaluate the same element $e * f$ using [A2] with $x = e$ : $e * f = e$ .

2.1

The two evaluations are of the same element of $S$ , so $e = e * f = f$ .

step 1.1step 1.2

3.1

If $e_{1}$ and $e_{2}$ are two-sided identities, then $e_{1}$ is in particular a left identity and $e_{2}$ a right identity, so $e_{1} = e_{2}$ by step 2.1; this is claim 1.

step 2.1given

3.2

If $e$ is a two-sided identity and $g$ is any left identity, then $g$ is a left identity and $e$ a right identity, so $g = e$ by step 2.1; if $g$ is any right identity, then $e$ is a left identity and $g$ a right identity, so $e = g$ by step 2.1. This is claim 2.

step 2.1given

4.1

Hence a left identity and a right identity always coincide, and a two-sided identity, when it exists, is unique and is the unique one-sided identity on either side.

step 2.1step 3.1step 3.2∎

Remarks

The argument is the whole content of the phrase "the identity": the element $e * f$ is read in two ways, and the two readings are forced to agree. Nothing beyond the two defining equations is used, so the lemma applies to any binary operation whatever, associative or not.
The same two-readings device, one level up, gives uniqueness of inverses (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided); there associativity is needed, and it is what makes the argument work.

Depends on

Used by

Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides Definition
Semigroup and monoid Definition
Vector space over a field Definition

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Identity element (Wikipedia) (standard reference, not scraped)