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Question

What is the difference between an "algebra" and an "algebraic structure"?

Where I'm at

I've done the wiki, google, and overflow search for definitions on "algebra" and "algebraic structure". My current findings are:

Algebra - "Study of mathematical symbols and rules governing their manipulation"

Algebraic Structure - “A set with a finite number of operations constrained by axioms”

Why

Trying to define monoids in terms of "algebra" and "algebraic structure" so that I can understand on a deeper level.

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3 Answers

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Algebra is the area of study, while an algebraic structure is the object being studying. In other words, algebra is the study of algebraic structures.

What might be confusing is that there are certain algebraic structures which are themselves called algebras.

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"Algebra," the subject, is the study of algebraic structures, such as groups, rings, fields, etc.

Confusingly, there is also a particular algebraic structure called an "algebra," which I think is probably what you're asking about. See:

A monoid is another algebraic structure.

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Algebra is only the territory, it also brief-language. An algebraic structure is a set with one or more operations in it.

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