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Across many of the algebras, including those of sets, groups, categories etc., it's often noted that the presence of an identity operation (or lack thereof) is a major trait in distinguishing between various abstractions; e.g. a monoid is distinct from a semigroup by virtue of having an identity operation.

Coming from a programming background, though, I'm having trouble grasping the significance of such a trivial operation; even as I work with Haskell, which takes some cues from category theory and includes monoid structures, I never find myself using id operations.

What are the uses and implications of having an identity operation over a certain abstraction?

Jules
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1 Answers1

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Not having an identity means there are no inverses for any element.

In general you can add an identity element to an algebraic structure: given a semigroup just declare 1 to be an element with the property 1x=x=x1 for all x. The semigroup axioms still hold.

Geva Yashfe
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