Algebraic Structures

Once we have defined our set we want to define the operations, normally binary operations, on the elements of such set. The operations we define must also satisfy a finite set of identities called axioms.

The combination of a nonempty set, the operations on the elements and the axioms for these operation is what we call algebric structure.

Binary operations

A binary operation on a set $S$ is a function

$$ \cdot : S \times S \to S $$

that takes two elements of $S$ and returns another element of $S$.

A fundamental requirement we will often impose is closure: applying the operation to elements of $S$ must always produce an element still in $S$.

Semigroups

A semigroup is a pair $(S, \cdot)$ where $S$ is a set and $\cdot$ is a binary operation such that the following properties hold:

Closure: for all $a, b \in S$, $$ a \cdot b \in S $$
Associativity: for all $a, b, c \in S$, $$ (a \cdot b) \cdot c = a \cdot (b \cdot c) $$

Associativity is the key structural property here.

For brevity, we will often omit the symbol $\cdot$ and write $ab$ instead of $a \cdot b$, whenever the meaning is clear from the context.

Two elements $a, b \in S$ are said to commute if $ab = ba$. A semigroup is called abelian (or commutative semigroup) if every $a, b \in S$ commute, such that:

$$ \forall a, b \in S, \quad ab = ba $$

The term abelian is more commonly used in the context of groups, but the notion of commutativity applies to any algebraic structure with a binary operation.

Cancellation Laws

Given a semigroup $(S, \cdot)$ and an element $x \in S$, we say that:

$x$ is left cancellable if, for all $a, b \in S$, $$ xa = xb \implies a = b $$
$x$ is right cancellable if, for all $a, b \in S$, $$ ax = bx \implies a = b $$

A semigroup in which every element is left and right cancellable is called a cancellative semigroup.

Not all semigroups satisfy cancellation. This is a strong additional property.

Powers in a Semigroup

Let $a \in S$, we define the powers of an element $a$ in the following way:

$$ a^1 = a $$

$$ \forall n \in \mathbb{N} \text{ with } n>1, \quad a^{n+1} = a^na $$

Because of associativity, expressions like $a \cdot a \cdot a$ do not depend on how we place parentheses, so $a^n$ is well-defined.

This is one of the first concrete consequences of associativity.

An element $x \in S$ is called idempotent if $x^2 = x$.

Example: In $(\mathbb{N}, +)$, the element $0$ is idempotent since $0 + 0 = 0$.

Monoids

A semigroup $(M, \cdot)$ is called a monoid if there exists $e \in M$ (called the identity) such that

$$ \forall a \in M \quad ea = ae = a $$

This element is unique, indeed let $e, e' \in M$ two identity elements, then

$$ e = ee' = e' $$

Because of this, we can denote a monoid by the triple $(M, \cdot, e)$.

Powers in a Monoid

The existence of the identity allows us to extend the definition of powers. For every $a \in M$, we define: $$ a^0 = e $$

Together with the recursive definition $$ a^{n+1} = a^na $$ this gives a consistent definition of $a^n$ for all $n \in \mathbb{N}$.

The identity element is idempotent, indeed

$$ e^2 = ee = e $$

Invertible elements

Let $(M, \cdot, e)$ be a monoid, an element $a \in M$ is called invertible if there exists an element $b \in M$ such that

$$ ab = ba = e $$

The element $b$ is called the inverse of a, and it is unique, indeed let $b, c \in M$ be two inverse of $a$, then

$$ b = be = b(ac) = (ba)c = ec = c $$

The uniqueness of the inverse of an element $a$ allow us to denote it by $a^{-1}$.

Groups

A group is a monoid $(G, \cdot, e)$ where every element has an inverse.

Formally, a group $(G, \cdot, e)$ satisfies:

Associativity of the $\cdot$ operation
Identity element $e$
For every $a \in G$, there exists $a^{-1} \in G$ such that $$ aa^{-1} = a^{-1}a = e $$

If the operation is also commutative, the group is called abelian, of course a group is abelian if the underlying semigroup is commutative.

We can now demonstrate that every group satisfies the cancellation law, indeed if $ab = ac$ then

$$ b = eb = (a^{-1}a)b = a^{-1}(ab) = a^{-1}(ac) = (a^{-1}a)c = ec = c $$

similarly, if $ba = ca$ then

$$ b = be = b(aa^{-1}) = (ba)a^{-1} = (ca)a^{-1} = c(aa^{-1}) = ce = c $$

Subgroups

Let $(G, \cdot, e)$ be a group, a subgroup is a subset $H \subseteq G \ne \emptyset$ that is itself a group under the same operation, such that:

$\forall a, b \in H$, $ab \in H$
$\forall a \in H$, $a^{-1} \in H$

It is clear that $e \in H$ because $\forall a \in H$, $aa^{-1} = e \in H$.

A subgroup $H \le G$ is called proper if $H \neq G$.

We write $H \le G$ to denote that $H$ is a subgroup of $G$, and $H < G$ if $H$ is a proper subgroup.

It is easy to verify that a nonempty subset $H \subseteq G$ is a subgroup if and only if $$ \forall a, b \in H,\ a^{-1}b \in H $$

Indeed, if $H \le G$, then for $a, b \in H$, we have $a^{-1} \in H$ and thus $a^{-1}b \in H$.

Conversely, suppose $H \neq \emptyset$ and $a^{-1}b \in H$ for all $a, b \in H$. Then:

Taking $a = b$, we get $e \in H$
Taking $b = e$, we get $a^{-1} \in H$
Then closure follows since $ab = (a^{-1})^{-1}b \in H$

Intersection of Subgroups

Let $(G, \cdot, e)$ be a group and let ${H_i}{i \in I}$ be a family of subgroups of $G$, then $$ H = \bigcap{i \in I} H_i $$

is also a subgroup of $G$.

Indeed, since $e \in H_i$ for all $i \in I$, we have $e \in H$. Moreover, if $x, y \in H$, then $x, y \in H_i$ for all $i \in I$, hence $x^{-1}y \in H_i$ for all $i \in I$, so $x^{-1}y \in H$.

Unions of Subgroups

Let $(G, \cdot, e)$ be a group, and let $H$ and $K$ be subgroups of $G$, then $$ H \cup K \text{ is a subgroup of } G \iff H \subseteq K \text{ or } K \subseteq H $$

This implies that a group cannot be written as the union of two proper subgroups.

Indeed, suppose by contradiction that there exist two proper subgroups $H < G$ and $K < G$ such that $$ G = H \cup K $$

Then by the previous statement, either $H \subseteq K$ or $K \subseteq H$. Suppose $H \subseteq K$, then $$ G = H \cup K = K $$ which contradicts the fact that $ K $ is a proper subgroup of $G$.

Rings

A ring is a triple $(R, +, \cdot)$ where $R$ is a set, $+$ is an binary operation $R$ and $\cdot$ is an associative binary operation on $R$ such that the following properties hold:

$(R, +, 0)$ is an abelian group
$(R, \cdot)$ is a semigroup
Distributive laws hold: $$ a(b + c) = ab + ac $$ $$ (a + b)c = ac + bc $$

If there exists an element $1 \in R$ such that $1 \ne 0$ and $\forall a \in R$, $1 \cdot a = a \cdot 1 = a$ then $A$ is called a unitary ring and $1$ is called the unity of the ring.

This element is unique, like we said in the groups.

Given two elements $a, b \in R$ we say that they commute if $ab = ba$. A ring is called commutative ring if $\forall a, b \in R$, $ab = ba$.

Units of a Ring

We now define the set of units of $R$, denoted by $U(R)$ that is the set of all elements in $R$ that are invertible with respect to the $\cdot$ operation:

$$ U(R) = {\forall a \in R | \exists b \in R : ab = ba = 1 } $$

The set $U(R)$ is a group under the $\cdot$ application

Subrings

Let $(R, +, \cdot)$ be a ring, a subring is a subset $S \subseteq R \ne \emptyset$ that is itself a ring with the same operations, such that

S is a subgroup of $(R, +, 0)$
$\forall a, b \in S$, $ab \in S$

In the case $(R, +, \cdot)$ is a unitary ring, then the following is required as well

$1 \in S$

Intersection of Subrings

Let $(R, +, \cdot)$ be a ring and let ${S_i}{i \in I}$ be a family of subrings of $R$, then $$ S = \bigcap{i \in I} S_i $$

is also a subring of $R$.

Indeed, $S$ is a subgroup of $(R, +, 0)$. For $a, b \in S$ then $\forall i \in I$, $a, b \in S_i$, then $\forall i \in I$, $ab \in S_i$. Thus $ab \in S$. This demonstrates that $S$ is a subring of $R$. If $R$ is a unitary ring, then $\forall i \in I$, $1 \in S_i$ and then $1 \in S$.

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