Turns out that having a page about a subject you're learning is a lovely hobby. Think of what you see below as an unfinished mathematical writing exercise made to hold things I'm studying in the "Algebra with Coffee" meetings that I attend every Thursday. There are some quirks that I'm still trying to figure out, like if I should abuse more logical notations or not, but hopefully you can read without much problem.

To start out, we should make some basic definitions about what we mean by an operation and then proceed to define what an algebraic structure is. Take \(A\) as a set and \(n\) a positive integer. We define \(A^n\) to be the set of all \(n\)-tuples of elements of \(A\), and \(A^0\ = {\varnothing}\).

For any \(A\) and \(n\) as above, we call a function \(A^n \rightarrow A\) an n-ary operation on \(A\). The natural number \(n\) is called the rank of the operation. Note that when \(n=0\) we have a function \(c : {\varnothing} \rightarrow A\) which is completely determined by the value \(c({\varnothing}) \in A\). Nullary operations or constants, as they can be called, are just the elements of A.

An algebra is a pair \(\langle A, F\rangle\) in which \(A\) is a nonempty set and \(F = \langle f_i \; | \; i \in I \rangle \) is a family of operations on \(A\), indexed by some set \(I.\)
The set \(A\) is called the universe of the algebra, and the \(f_i\) are the fundamental or basic operations.

Let \(A = \langle A, F \rangle\) be an algebra with \(F = \langle f_i \; | \; i \in I\rangle. \) The function \(\rho : I \rightarrow \mathbb{N}\) which assigns to each \(i \in I\) the rank of \(f_i\) is called the similarity type of \(A.\) Two algebras are called similar if they have the same similarity type.

Take \(\textbf{A} = \langle A, F \rangle\) and \(\textbf{B} = \langle B, G \rangle\) to be algebras of the same similarity type \(\rho : I \rightarrow \mathbb{N}\). We call \(\textbf{B}\) a subalgebra of \(\textbf{A}\) if \(B \subseteq A\) and \(\forall i \in I\), \(g_i = f_i \upharpoonright B^{\rho(i)}.\) The notation may seem confusing, but we are just essentially stating that any operation of \(G\) can be considered as an operation of \(F\) with the domain resctriction of \(B^{\rho(i)}\), where \(\rho(i)\) is the arity of the operation. To say that \(B\) is a subalgebra of \(A\) we denote \(B \leq A\).

A function \(h: B \rightarrow A\) is called a homomorphism if for every \(i \in I\) and every \(b_1,b_2,...,b_n \in B\), \(h(g_i(b_1,...,b_n)) = f_i(h(b_1),...,h(b_n)).\) \((n = \rho(i)\)).
The thing that makes \(h\) "special" is its ability to preserve operations, and we shall see many uses of functions with such a property.

This first example will be useful to define the next one. A monoid is an agelbra \(\langle M, \cdot, e \rangle\) statisfying the associative law and the identities \(x \cdot e = x\) and \(e \cdot x = x\). Note that "\(\cdot\)" is a binary operation and "\(e\)" a nullary operation.

A group is an algebra \(\langle G, \cdot, ^{-1}, e \rangle\) of similarity type \(\langle 2,1,0 \rangle\), such that \(\langle G, \cdot, e \rangle\) is a monoid and satisfying \(x \cdot x^{-1} = x^{-1} \cdot x = e\). If it is also true that for any two elements of \(G,\) lets say \(x\) and \(y,\) \(x \cdot y = y \cdot x,\) then \(G\) is called an Abelian group.

There are many other examples that can be given, like rings and such (you could even create your own algebra), but these two should be enough.

To make things easier, especially in notation, since most of times when we talk about several algebras in the same phrase they have the same similarity type, we will fix a similarity type and then consider algebras of that type.
Geez! I repeated the word "type" too many times.

Therefore, instead of I, lets take for an index set a family \(\Omega \) of operation symbols, and let \(\rho : \Omega \rightarrow \mathbb{N}\). Then by an algebra \(\textbf{A}\) of type \(\rho\) we mean a set \(A\) together with, for each operation symbol \(\omega \in \Omega\), an operation \(\omega^{\textbf{A}}\) of rank \(\rho(\omega)\). Thus \(\textbf{A} = \langle A, \Omega^{\textbf{A}} \rangle \) where \(\Omega^{\textbf{A}} = \langle \omega^{\textbf{A}} \; | \; \omega \in \Omega \rangle\). Thankfully, we will be able to omit the superscript most of the time without being confused if we're talking about the operation symbol or the operation itself.

Proposition. Let \(\textbf{A}\) be any algebra, and let \(\mathcal{S}\) be any nonempty collection of subalgebras of \(A.\) Then \(\bigcap\limits_{B \in \mathcal{S}} B \leq \textbf{A}\).

Let \(\textbf{A}\) be an algebra and \(X \subseteq A.\) Consider the set \(\mathcal{B}_X = \langle B \leq A \; | \; X \subseteq B \rangle\) a family of subalgebras of \(\textbf{A}\) that contain \(X\). The minimal subalgebra generated by \(X\) is the minimal subalgebra in the set \(\mathcal{B}_X.\) We denote \(\langle X \rangle.\)

Proposition. \(\langle X \rangle = \bigcap\limits_{B \in \mathcal{B}_X} B.\)

To get a more clear idea of what we mean by \(\langle X \rangle\) we can use the following proposition.

Proposition. Let \(\textbf{A} = \langle A, \Omega \rangle\) be an algebra and \(X \subseteq A.\) Define, by recursion on n, the sets \(X_n\) by:

\(X_0 = X\);
...
\(X_{n+1} = X_n \cup \{ \omega(x_1,...,x_k) \; | \; x_1,...,x_k \in X_n\), \(\omega \in \Omega\), and \(k = \rho(\omega)\}\).
Then \(\langle X \rangle = \bigcup\limits_{n = 0}^{\infty} X_n.\)

There are a few handful definitions regarding homomorphisms we might make. Let \(\phi : \textbf{A} \rightarrow \textbf{B}\) be a homomorphism.
If \(\phi\) is an injective (i.e., one-to-one) homomorphism, we can call \(\phi\) an embedding or a monomorphism.
If \(\phi\) is a surjective homomorphism, then \(\phi\) is an epimorphism and \(\textbf{B}\) is called a homomorphic image of \(\textbf{A}\).
If \(\phi\) is a bijective homomorphism, then \(\phi\) is an isomorphism and the algebras \(\textbf{A}\) and \(\textbf{B}\) are called isomorphic.
If \(\phi\) is a homomorphism from \(\textbf{A}\) to \(\textbf{A}\), then \(\phi\) is an endomorphism.
If \(\phi\) is an endomorphism and an isomorphism, then \(\phi\) is an automorphism.
That's about it, I hope you aren't feeling sleepy .

Proposition. Let \(\phi : A \rightarrow B\) and \(\psi : B \rightarrow C\) be two homomorphisms. Then \(\psi \circ \phi : A \rightarrow C\) is a homomorphism.

Let \(\phi : A \rightarrow B\) be a homomorphism. We define \(ker\phi = \{(a_1,a_2) \in A^2 \; | \; \phi(a_1) = \phi(a_2)\}\).

Proposition. Take \(\phi : A \rightarrow B\) to be a homomorphism. If \(A = \langle X \rangle\), then \(im\phi = \langle \phi(X) \rangle.\)