An algebraic system is simply a set, equipped with one or more operations. The number and nature of the associated operations are part of the characteristics of the system, so the system of integers under addition is distinct from the system of integers under multiplication.
The operations may be unary, binary, etc. Hence, they all have $S^{n}, n \geq 1$ as their domain, where $S$ is the set associated with the system. When the operations satisfy certain axioms (properties), the system is called an algebraic structure.
In order to specify a system, therefore, we need to specify both the set as well as the operations. Hence, a system with $n$ operations is represented as an $(n+1)$-tuple, where the first element is the set, and the rest are the operations. For example, the two systems referred to above are denoted by $\langle \mathbb{Z}, + \rangle$ and $\langle \mathbb{Z}, \cdot \rangle$ (or $\langle \mathbb{Z}, \times \rangle$) respectively.
Abstract algebra is the study of the properties of such sets under such operations.
Closure (sets). When the output of an operation acting on a set is always an element of the set, the set is said to be closed under the operation. For example, adding two integers together always gives an integer; hence the integers are closed under addition. However, dividing one integer by another may not yield an integer, so the integers are not closed under division.
Commutativity (operations). When the order of operands does not matter, the operation is said to be commutative in the set. For example, addition and multiplication are commutative in $\mathbb{R}$, but subtraction and division are not.
Associativity (binary operations). When three elements in a fixed order give the same result no matter which two are combined first, the operation is said to be associative in the set, i.e., bracketing makes no difference. For example, $(a+b)+c$ = $a+(b+c)$, $\forall a,b,c \in \mathbb{R}$, so addition is associative in $\mathbb{R}$. However, in general, $(a-b)-c \neq a-(b-c)$, so subtraction is not associative in $\mathbb{R}$.
Cancellativity (sets/elements). When an element can be cancelled in an equation (as we are familiar with from multiplication in real numbers), it is said to be cancellative. If the operation is not commutative, left and right cancellativity can be distinguished. An element $a \in S$ is left-cancellative if $a \cdot b = a \cdot c \Rightarrow b = c$, $\forall b,c \in S$, where $\cdot$ is the operation. Similarly, $a$ is right-cancellative if $b \cdot a = c \cdot a \Rightarrow b = c$, $\forall b,c \in S$. To say that an element is cancellative is to say that it is both left- and right-cancellative.
If a set is (left- or right-) cancellative, it means that all elements of the set are (left- or right-) cancellative. Note that (left- or right-) cancellativity of a closed set $S$ is equivalent to saying that the equations $a \cdot x = b$ and $y \cdot a = b$ have unique (but not necessarily equal) solutions $x,y$, $\forall a,b \in S$. This is proved below for left-cancellativity; the proof for right-cancellativity is analogous.
(i) Left-cancellativity $\Rightarrow$ unique solution
Consider the function $f_{a}: S \rightarrow S$, defined as $f_{a}(x) = a \cdot x$. If it is invertible, then the equation would have a unique solution. Its range is:
By closure, $S_{a_{L}} \subseteq S$. If $|S_{a_{L}}| < |S|$, then (by PHP), $a \cdot x_{1} = a \cdot x_{2}, x_{1} \neq x_{2}$ for some $x_{1},x_{2} \in S$ (i.e., $f_{a}$ is many-one). However, $a$ is left-cancellative; hence $x_{1} = x_{2}$ and $f_{a}$ is one-one. This is a contradiction – hence $|S_{a_{L}}| = |S|$ and $S_{a_{L}} = S$.
If $f_{a}$ were not onto, then (again by PHP), $a \cdot x_{1} = a \cdot x_{2}, x_{1} \neq x_{2}$. We know this to be false, so $f_{a}$ is surjective also. Therefore, $f_{a}$ is bijective and invertible, and the equation does have a unique solution.
(ii) Unique solution $\Rightarrow$ left-cancellativity
If the equation has a unique solution, $f_{a}$ as defined is invertible, and therefore injective. Injectivity implies left-cancellativity; this completes the proof.
Note: The crucial step in (i) was the proof of surjectivity. The reverse implication does not require this; hence it was trivial.
Existence of Identity (sets). An identity element is one that does not change any other when it is operated with it. In $\mathbb{R}$, 0 is the additive identity $(x+0=x$, $\forall x \in \mathbb{R})$ and 1 the multiplicative identity $(x \cdot 1 = x$, $\forall x \in \mathbb{R})$.
Existence of Inverse (sets/elements). The inverse of an element is one which, when operated with, yields the identity element. If the operation is not commutative, left and right inverses can be distinguished. The left inverse of an element $a$ would be an element $l$ such that $l \cdot a = e$, $e$ being the identity element and $\cdot$ the operation. Similarly, the right inverse of $a$ would be an element $r$ such that $a \cdot r = e$. Note that this property is equivalent to (left- or right-) cancellativity.
| Name | Property/Properties | Example | Counterexample |
|---|---|---|---|
| Groupoid/Magma | Closure | $\langle \mathbb{N}, + \rangle$ | $\langle \mathbb{Z}, \div \rangle$ |
| Semigroup | Closure, Associativity | $\langle \mathbb{Z}, \cdot \rangle$ | $\langle \mathbb{N}, - \rangle$ |
| Monoid | Closure, Associativity, E. of Identity | $\langle \mathbb{Z}, \cdot \rangle, e = 1$ | $\langle \mathbb{Z}_ {n}^{+}, +_{n} \rangle$ |
| Quasigroup | Closure, Cancellativity | $\langle \mathbb{Z}, - \rangle$ | $\langle \mathbb{V}_ {2}, \times \rangle$1 |
| Group | Closure, Associativity, E. of Inverse, E. of Identity | $\langle \mathbb{Z}_ {n}, \times_{n} \rangle$ | $\langle \mathbb{N}, \div \rangle$ |
Note that (i) all semigroups, monoids, quasigroups and groups are groupoids, (ii) all monoids and groups are semigroups, and (iii) all groups are groupoids, semigroups, quasigroups and monoids.
A semigroup or monoid is said to be cyclic if all its elements can be obtained from one of its elements by repeated application of the operation. Formally, for a system $\langle S, \cdot \rangle$, $\exists s \in S$ such that S can be written as
where $s^{0}$ is defined as the identity element $e$, $\forall s \in S$. The element $s$ is called a generator of $S$ (there may be more than one generator). Note that this property makes no reference to the finiteness of $S$; however the terminology is unintuitive (but applicable) for infinite systems as well.
A group whose operation is commutative is called an abelian group.
$\mathbb{V}_{2}$ = set of all vectors in the plane (non-standard notation); $\times$ = cross product ↩