📂Abstract Algebra

Semigroup in Abstract Algebra

Definition ¹

For an element $a,b,c$ of magma $\left< S, *\right>$, if $(a \ast\ b) \ast\ c = a \ast\ (b \ast\ c)$, then $\left< S, *\right>$ is defined as a Semigroup.

Explanation

A semigroup is a magma where the operation satisfies the associative law.

It is not necessary to have an identity element or an inverse element, only the associative law needs to be satisfied. Regardless of how easy it is to prove the associative law, it is quite natural that the discussion goes to the associative law right after closure. Having an operation but no associative law? By this point, it’s difficult to find examples that have some algebraic significance and do not become semigroups.

Let’s look at an example of something that becomes a magma but not a semigroup.

For the set $S = \left\{ a,b,c \right\}$, the magma $\left< \mathscr{P}(S) , \setminus \right>$ is not a semigroup.

• Because $$( \left\{ a,b,c \right\} \setminus \left\{ a \right\} ) \setminus \left\{ b \right\} = \left\{ b,c \right\} \setminus \left\{ b \right\} = \left\{ c \right\} $$
  and
  $$ \left\{ a,b,c\right\} \setminus ( \left\{ a \right\} \setminus \left\{ b \right\} ) = \left\{ a,b,c\right\} \setminus \left\{ a \right\} = \left\{ b, c \right\}$$
  thus
  $$( \left\{ a,b,c\right\} \setminus \left\{ a \right\} ) \setminus \left\{ b \right\} \ne \left\{ a,b,c\right\} \setminus ( \left\{ a \right\} \setminus \left\{ b \right\} )$$

As you can see, the example is quite peculiar. It’s not common to find such easily understandable examples outside of dealing with algebraic structures. On the other hand, it is very easy to find examples of semigroups.

Magma $\left< \mathbb{N} , + \right>$ is a semigroup.

• Since adding natural numbers together results in a natural number, $\left< \mathbb{N} , + \right>$ is a magma. Since the order of adding natural numbers does not affect the sum, $\left< \mathbb{N} , + \right>$ becomes a semigroup.