Operations

Definition

An operation $*$ on any set $A$ is a rule which assigns to each ordered pair $(a, b)$ of elements of $A$ exactly one element $a * b$ in $A$.

1. $a * b$ is uniquely defined for every ordered pair $(a, b)$ of elements of $A$.

2. $a * b$ must be uniquely defined.

3. $A$ must be closed under the operation $*$.

There are, in general, many possible operations on a given set $A$. A set with two distinct elements has $2^4 = 16$ possible operations.

Properties of Operations

An operation may be:

1. Commutative: $\forall a, b \in A: a * b = b * a$.
2. Associative: $\forall a, b, c \in A: (a * b) * c = a * (b * c)$.

Identity

There may exist an identity or neutral element w.r.t the operation $*$.

How is identity element different from identity morphism? What is the relationship between operation in abstract algebra adn morphism in category theory

• Left Identity: $\forall a, e \in A: e * a = a$.
• Right Identity: $\forall a, e \in A: a * e = a$.
• Two-sided Identity (or simply Identity): $\forall a, e \in A: e * a = a \wedge a * e = a$.

There are a few possibilities for existence of an identity element.

• Having no identity element at all.
• Having several left identities.
• Having several right identities
• Having a single two-sided identity.

Inverse

There may exist an inverse element w.r.t the operation $*$.

math \forall a, b, e !\in! A ! : a * b = e

$a$ is the left inverse of $b$ and $b$ is left invertible. A left-invertible element is left-cancellative. Similarly, $b$ is the right inverse of $a$ and $a$ is right invertible. A right-invertible element is right-cancellative.

math \forall a, x, e !\in! A ! : a * x = e \land x * a = e

$x$ is a two-sided inverse or simply an inverse of $a$ and $a$ is invertible in $A$.

There are a few possibilities for existence of an inverse element.

• Having no inverse element at all.
• Having several left inverses.
• Having several right inverses
• Having a single two-sided inverse.

Cancellation Property

The notion of cancellation is a generalization of the notion of invertible.

Left Cancellation Property (left-cancellative):

math \exists a !\in! (M, *) \colon \forall b, c \in M \colon a * b = a * c \implies b = c

Right Cancellation Property (right-cancellative):

math \exists a !\in! (M, *) \colon \forall b, c \in M \colon b * a = c * a \implies b = c

Absorbing / Annihilating Element

There may exist an absorbing or annihilating element w.r.t binary operation $*$. This is also called a zero element.

math z \in (S,*) \colon \forall s \in S \colon z * s = s * z = z

Left zero:

math \exists a !\in! (S, *) \colon \forall s \in S \colon z * s = z

Right zero:

math \exists a !\in! (S, *) \colon \forall s \in S \colon s * z = z

If a magma has a zero element, then it is unique.