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I am slightly confused about the definition of an algebra. Depending on the context the definitions seem to vary widely. For example this source in the context of measure theory defines an algebra as follows:

A collection $\mathcal{A}$ of subsets of $X$ is called an algebra if:

  1. $\emptyset \in \mathcal{A}$,
  2. if $A,B \in \mathcal{A}$ then $A\cap B\in \mathcal{A}$;
  3. if $A\in \mathcal{A}$ then $A^c\in \mathcal{A}$.

whilst (Dummit and Foote, 2004;pg342) define an algebra as follows:

Let $R$ be a commutative ring with identity. An $R$-algebra is a ring $A$ with identity together with a ring homomorphism $f:R\rightarrow A$ mapping $1_R$ to $1_A$ such that the subring $f(R)$ of $A$ is contained in the center of $A$.

Clearly these two examples are very different. As such is there a overall definition of an algebra which includes all such definitions (i.e. if $A$ is an algebra under one of these then it will be an algebra under the overall definition)? If not does anyone know of any source that lists the definitions of 'an algebra' in different contexts?

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  • I don't know about the origin of the term, but certainly the first one forms a ring with symmetric difference and intersection ( https://math.stackexchange.com/questions/1986529/power-set-of-x-is-a-ring-with-symmetric-difference-and-intersection ) and hence, an algebra over itself. – Severin Schraven Aug 27 '17 at 16:24
  • At first a $k$-algebra $A$ is what you need to consider polynomials $\sum_{n=0}^d c_n a^n$ with coefficients $c_n \in k$ and $a \in A$. An algebra of sets becomes a boolean algebra (algebra over $\mathbb{F}_2$) when you consider the addition $a \oplus b = { x \in a, x \not \in b} \cup {x \in b, x \not\in a}$ (termwise xor). The union of sets $a \cup b$ is not an addition, but infinite union and intersection have certain good properties, allowing some sort of completions like $\sigma$-algebras. – reuns Aug 28 '17 at 09:25

1 Answers1

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General Definition of an Algebra

In general we can define an algebra as follows (from {1;pg2}, verbatim):

An algebra is an ordered pair $\langle A,F\rangle$ such that $A$ is a nonempty set and $F=\langle F_i: i \in I\rangle$ where $F_i$ is a finitary operation on $A$ for each $i\in I$. $A$ is called the universe of $\langle A, F\rangle$, $F_i$ is refereed to as a fundamental or basic operation of $\langle A,F \rangle$ for each $i \in I$, and $I$ is called the index set or the set of operation symbols of $\langle A,F\rangle$.

(emphasis not mine).

Terminology

Definition 1 above is actually defining an algebra of sets rather then a general algebra whilst definition 2 is defining an $R$-algebra where $R$ is a ring.

References

{1} http://www.math.hawaii.edu/~ralph/Classes/619/ALVin.pdf

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