Are all algebras groups?

Question

It seemed to me that boolean algebra is a group because it is closed (You can't use boolean algebra and get a result that is outside the group) under a logical primitive(?) and order of operands and order of operators don't matter (0+1=1+0). But is it even defined what we mean when we say "an algebra" (abstract algebra, boolean algebra, matrix algebra, Lie algebra...?) and can we know that any "algebra" also is a group?

Answer 1

There's a couple things going on here.

First, Boolean algebras are not groups in any immediate sense. A group is a set with a binary operation which is associative, has an identity, and has inverses; neither of the main binary operations of a Boolean algebra satisfy this.

That said, we can turn a Boolean algebra into a group with some work: if $(B; \wedge, \vee, \neg)$ is a Boolean algebra, then $(B; XOR)$ is a group - "exclusive or" is associative and has an identity ($\bot$) and inverses (each element is its own inverse).

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I think what you ultimately want is universal algebra.

An algebra is just a set $X$ with a family of functions $\mathcal{F}$ on (finite Cartesian powers of) $X$. We also say $(X, \mathcal{F})$ is a $\Sigma$-algebra, where $\Sigma$ is a set of function symbols, if there's a bijection ("interpretation") between $\Sigma$ and $\mathcal{F}$ which preserves arity (an $n$-ary function symbol gets mapped to an $n$-ary function).

Examples:

• A group is a $(*, ^{-1}, e)$-algebra, where $*$ is a binary function symbol, $^{-1}$ is a unary function symbol, and $e$ is a zero-ary function symbol (= constant symbol).

• A Boolean algebra is a $(\wedge, \vee, \neg, \top, \bot)$-algebra, where $\wedge$ and $\vee$ are binary function symbols, $\neg$ is a unary function symbol, and $\top$ and $\bot$ are zero-ary function symbols (= constant symbols).

(Of course, that's not all a group or Boolean algebra is - we also demand that they satisfy some equations in the relevant language.)

Now, from any set of function symbols, we can form the set of polynomials over that set: basically, all the functions built up from those function symbols and variable symbols. E.g. in the language of Boolean algebras, the following are polynomials:

• $y\mapsto y\wedge y$,

• $x, y\mapsto x\vee (x\wedge y)$

• $x, y\mapsto (x\vee y)\wedge (\neg x\vee \neg y)$ (this one is "XOR").

What we've observed above is that there are terms in the language of Boolean algebras, which turn the set into a group. In a particularly nice sense, every Boolean algebra "interprets" a group (there is a very wide class of notions of "interpret" here).

Universal algebra, among other things, studies the structural properties of algebras in terms of their polynomials. I think this would be interesting to you.

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EDIT: You should note that "algebra" often refers to something much more specific - the "ring version" of a vector space (https://en.wikipedia.org/wiki/Algebra_over_a_field). But I don't think that's what you're looking for here.