Let $R$ be a ring.
Semi-group Algebra
Let $M$ be a semi-group, $f : M \to R$. we write $f = \sum f_m x^m$ formally and call $f_m \in R$ the coefficient of $f$ at $x^m$, where $f_m$ is the image of $m$ in $R$ under $f$.
Let \(R[[M]] = \\\{f \mid f : M \to R\\\}\) the algebra of formal series, and \(R[M] = \\\{f \mid f : M \to R, \\\# \\\{m \mid f(m) = 0 \\\} < +\infty \\\}\) the algebra of polynomials.
Notice that $R[[M]]$ is the free $R$-module with basis $M$. $R[M]$ are those elements with finite support.
Define a binary operation on $R[[M]]$ \(fg = \sum (\sum\_{hk = m} f\_h g_k) x^m\) It’s easy to verify that this operation is associative.
Examples
Now take $G = (\mathbb N, +)$, the semi-ring of natural numbers, with addition operation, then $R[G]$ is right the polynomial ring of one variable, we usually denoted by $R[X_1]$. The formal sum $\sum f_m x^m$ now becomes the generating function of the coefficient sequence.
When $G = \mathbb Z$, $R[G]$ becomes the Laurrent series.
When $G = (\mathbb N, \ast)$, $R[G]$ becomes the algrbra of arithmetic functions, and the product is the Dirichlet convolution.
Semi-groupoid Algebra
But the incidence algebra can not be defined in this way, which needs further genralization.
Recall the definition of incidence algebra: Let $P$ be a poset, $Q = \{(g, h) \mid g \leq h\} \subset P \times P$, $\alpha, \beta : Q \to R$, \(\alpha\beta = \sum (\sum\_{g\leq k \leq h} \alpha\_{(g,k)} \beta\_{(k,h)})x^{(g,h)}\)
We can define an associative partial binary operation $\circ$ on $Q$, \(\circ : Q \times Q \to Q\) \((x, y) \circ (y, z) \to (x, z)\) It doesn’t make $Q$ into a semi-group, this kind of algebra struture is called a semi-groupoid. A semi-group is the special case of a semi-groupoid.
Note that the operation for $\alpha\beta$ above is still a total binary operation on $R[Q]$. therefore we may generalize the definition of semi-group algebra to semi-groupoid algebra and still use the notation $R[Q]$ for the incidence algebra of $P$.
$R$-Algebra
Both semi-group algebra and semi-groupoid algebra are just $R$-algebras, where $R$ is a ring. The general definition is, an $R$-algebra $A$ is a ring and also an $R$-module. Which means, we have associative operations $A \times A \to A$ and $R \times A \to A$, and the operation $A \times A \to A$ is $R$-bilinear.
The set of square matrics of order $n$, $M_n(R)$ is a classical example of $R$-algebra. It can be view as $R[I_n^2]$, where $I_n^2 = \{(i, j) \mid 1 \leq i, j \leq n \}$. We will find that $I_n^2$ also admits a semi-groupoid structure.
First write a matrix as $a = \sum a_{ij} x^{ij}$, then the multiplication \(c\_{ij} = \sum a\_{ik}b\_{kj}\) can be rewrite as \(\sum c\_{ij}x^{ij} = \sum (\sum a\_{ik}b\_{kj}) x^{ij}\)
Now it’s clear that the partial binary operation on $I_n^2$ is \((i, k) \circ (k, j) \to (i, j)\) Which looks exactly the same as the operation on $Q$.
Think of $I_n^2$ and $Q$ are edges of some graphs. $I_n^2$ is the set of edges of a clique, $Q$ is the set of edges of a digraph. For edge $(x, y)$ and $(y, z)$, their connection is just $(x, z)$. Two edges can be connected if and only if the first edges ends at where the seconds edge starts.
This leads to the concept of path algebra.