Algebraic Structures

A (mathematical) structure is a system formalized as a set provided with some things.
An algebraic structure is a structure where those things are operators and axioms (and the result describes “an algebra”).
The “math” we learn in grade school is of a specific algebraic structure. There are unlimited other algebras all of which “math” can be done over.

An algebraic structure is a tuple $(\boldsymbol{A}, \text{op}_1, \text{op}_2, \dots, \text{ax}_1, \text{ax}_2, \dots)$ :

$\boldsymbol{A}$ – the Carrier Set
a nonempty set, the “objects” of this algebra
$\text{op}$ – operators
$\text{ax}$ – some (finite) axioms (or identities).

Here, an operator is a representative symbol of an operation.¹

Operations are finitary and closed (but not necessarily total).
In functional terms, they are endofunctions of finite arity, though they may be partial:
$\text{op}: \boldsymbol{A}^n \rightharpoonup \boldsymbol{A}, \quad n \in \N$

The axioms govern the operations.

In universal algebra:

An algebraic structure is (defines) an algebra;
The collection of all structures of a given type – same operators and same axioms – is called a variety

Common Axioms

Equational Axioms

Axioms that can be written as identities/equations. These relate directly with operations.
Observe that these are mainly for binary operations.

Commutativity
Associativity
Distributivity (Left- and/or Right-)
Idempotence
$x \ast x \equiv x$
(for some binary operator $\ast$ )
Absorption
$x \vee (x \wedge y) \equiv x \wedge (x \vee y) \equiv x$
(for some binary operators $\vee$ and $\wedge$ )

Existential Axioms

Axioms that are in some form of “there exists…” These relate with the contents of $\boldsymbol{A}$ .
These axioms can generally be substituted for further operators paired with equational axioms(s).

Identity (element $e$ , or 0-ary op $\text{id}$ )
$\exists e \,\forall x :\quad e \ast x = x \ \wedge\ x \ast e = x$
(for a binary operator $\ast$ )
Invertibility (elements $x^{-1}$ , or unary op $-$ )
$\forall x \,\exists \,x^{-1}=-x :\quad x^{-1} \ast x = e \ \wedge\ x \ast x^{-1} = e$
(for a binary operator $\ast$ )
Complement (elements $x'$ , or unary op $\neg$ )
$\forall x \,\exists x'=\neg x :\quad x \ \text{join}\ x' = \top \ \wedge\ x \ \text{meet}\ x' = \bot$
(for binary operators $\text{join}$ and $\text{meet}$ and elements $\top, \bot \in \boldsymbol{A}$ )

Observe that 2. is mainly for Group-like structures; 3. is for lattice-like structures.

Non-equational Axioms

Axioms can actually be any first-order formula.

Invertibility while also considering 0 (where $\forall x : x \ast 0 = 0$ )
$\forall x :\quad x=0 \ \vee\ x \ast x^{-1} = e$
Total(ity)
where an operator is well-defined for all elements of $\boldsymbol{A}$

When Invertibility is in conjunction with Identity and Associativity (e.g. in Groups), inverse elements are necessarily unique.
Complements are also unique in certain lattice structures.
Then, from uniqueness, it can be proved that the inverse/complement operations are involutions (self-inverses).

Common Types

These are categorized by their operators and axioms – type. (The contents of the carrier set $\boldsymbol{A}$ is not considered here.)

One set

Simple – 0 binary ops

$(\boldsymbol{A})$

Set
- degenerate algebraic structure with no operations (and axioms)
Pointed set
- $\boldsymbol{A} = (X, x_0, \dots)$
- set with “special” element(s) $x_0$ (“root”)
Set with unary op

Magma – 1 binary op, $+$ is total²

$(\boldsymbol{A}, +)$

Let $+$ represent the binary operator.
Commonly called add or “the group operation”.

Semigroup
- $+$ is associative
Monoid
- Semigroup properties, and
- identity
Group
- Monoid properties, and
- invertibility
Abelian Group / Commutative Group
- Group properties, and
- $+$ also commutes
Semilattice
- Semigroup properties, and
- $+$ also commutes
- $+$ is idempotent
- group operation usually called “meet” ( $\wedge$ ) or “join” ( $\vee$ ) instead of “add” ( $+$ )
- i.e., is “half” of Lattice below

Ringoid – 2 binary ops, $\ast$ distributes over $+$

$(\boldsymbol{A}, +, \ast)$

Let $+$ and $\ast$ represent the binary operators.
Commonly called add and multiply.

Ring
- $(\boldsymbol{A}, +)$ is an Abelian Group
- $(\boldsymbol{A}, \ast)$ is a Monoid
Division Ring / Skew Field
- Ring properties, and
- $(\boldsymbol{A}, \ast)$ is a Group ( $\ast$ has an inverse)³
Commutative Ring
- Ring properties, and
- $\ast$ also commutes
Field
- a Commutative Division Ring; i.e.,
- $(\boldsymbol{A}, +)$ and $(\boldsymbol{A}, \ast)$ are Abelian Groups³

Note that Additive Group and Multiplicative Group are names for the substructures of a Ringoid (the group-likes when considering only one of the two ring operations).

Lattice-like – 2+ binary ops, absorption

$(\boldsymbol{A}, \vee, \wedge)$

Let $\vee$ and $\wedge$ represent the binary operators.
Commonly called join and meet.

Lattice
- $\vee$ and $\wedge$ are idempotent, commutative, associative,
  and total
Complete Lattice
- Lattice properties, and
- $\vee$ and $\wedge$ are also total against subsets of $\boldsymbol{A}$ ⁴
Distributive Lattice
- Lattice properties, and
- $\vee$ and $\wedge$ are distributive over each other
Bounded Lattice
- Lattice properties, and
- has a $\top$ , greatest element in $\boldsymbol{A}$
- has a $\bot$ , least element in $\boldsymbol{A}$
- i.e., $\forall x \quad \bot \le x \le \top$
Complemented Lattice
- Bounded Lattice properties, and
- complements
Boolean Lattice / Boolean Algebra 😀
- a Complemented Distributive Lattice
Complete Boolean Algebra
- a Complete Boolean Lattice

Any complements in a distributive lattice are unique; Boolean Algebras have unique complements.

Completely Distributive Complete Boolean Algebras / Complete Atomic Boolean Algebras (CABAs) are isomorphic to power sets.⁵

Two(+) sets

kinda when some operators operate on more than one set (e.g., (in a module) when the carrier set $A$ 's is a superset of another (simpler) structure $B$ , and some operation(s) is of the form $\operatorname{op} : A \times B \mapsto A$ )
For this example, $\operatorname{op}$ is effectively partial, as it is only defined for a subset of $A$ as its second argument.

Module
- $(M, R, +, \times, \,\bm\cdot\,)$
- $R$ is a Ring (along its closed operations)
- $(M, +)$ is an Abelian Group ( $+$ is defined for $M$ as well as $R$ )
- scalar multiplication operation $\,\bm\cdot\,: R \times M \mapsto M$
  - left- and right- distributivity of $\,\bm\cdot\,$ over $+$ ; $x \in A$ and $r \in R$ distribute wrt. $\,\bm\cdot\,$ over each other’s sums⁶
  - associativity (with $R$ 's $\ast$ )⁶
  - identity⁶
Vector Space
- Module where $R$ is a Field
Algebra over a Field
- :exploding_head:
Inner product space
- :exploding_head:

In a Vector Space $(V, F, +, \times, \,\bm\cdot\,)$ , elements of $F$ are called scalars and vectors for those of $V$ (that is not in $F$ , should $V$ be considered a superset of $F$ ).

Uncommon Types

“Hybrid structures”
… structures with contents of non-algebraic nature(???)

I am clarifying this because operator is sometimes quite different from operation. ↩︎
there are “group-like” structures where $+$ is partial (though uncommon), e.g., Partial Magma ↩︎
$\ast$ has an inverse only for non-zero elements ↩︎ ↩︎
This matters when $\boldsymbol{A}$ is infinite; while totality on $\vee$ and $\wedge$ implies (via induction) totality against subsets, these subsets must be finite as induction does not “reach” infinity. We therefore need an amended axiom in the case of infinite subsets possible via an infinite $\boldsymbol{A}$ . ↩︎
assuming the principle of the excluded middle ↩︎
I.e., for $x, y \in M$ and $r, s \in R$ ,
1. $r \cdot (x + y) = r \cdot x + r \cdot y$
2. $(r + s) \cdot x = r \cdot x + s \cdot x$
3. $(r \times s) \cdot x = r \cdot (s \cdot x)$
4. $e \cdot x = x$ , where $e$ is $R$ 's $\times$ -identity element
Also, this formulation of rule 3 produces a left-module (of $R$ ); $(r \times s) \cdot x = s \cdot (r \cdot x)$ produces a right-module. ↩︎ ↩︎ ↩︎