Special classes of semigroups
A semigroup is like a club where people have a special way of combining things together. There are some semigroups that are extra special because they have some special rules.
One special type of semigroup is called a monoid. In a monoid, there is one member called the identity element. This member is special because when you combine it with anything else, it doesn't change the answer. It's like the number zero in addition – no matter what number you add zero to, you always get the same number back.
Another special type of semigroup is called a group. In a group, every member has an opposite member called an inverse element. When you combine a member with its inverse element, you get the identity element. It's like if you have five apples and you take away five apples, you end up with zero apples – the addition and subtraction cancel each other out.
A third special type of semigroup is called a regular semigroup. In a regular semigroup, every member has a special property where you can combine it with itself in different ways to get the same answer. This is like if you fold a piece of paper in half, then fold it again in half, you end up with the same result as if you had folded it once more.
These special classes of semigroups are important because they help us understand different ways that things can be combined together. Just like how we have different rules for adding, subtracting, multiplying and dividing numbers, these special semigroups have their own unique way of combining things.
One special type of semigroup is called a monoid. In a monoid, there is one member called the identity element. This member is special because when you combine it with anything else, it doesn't change the answer. It's like the number zero in addition – no matter what number you add zero to, you always get the same number back.
Another special type of semigroup is called a group. In a group, every member has an opposite member called an inverse element. When you combine a member with its inverse element, you get the identity element. It's like if you have five apples and you take away five apples, you end up with zero apples – the addition and subtraction cancel each other out.
A third special type of semigroup is called a regular semigroup. In a regular semigroup, every member has a special property where you can combine it with itself in different ways to get the same answer. This is like if you fold a piece of paper in half, then fold it again in half, you end up with the same result as if you had folded it once more.
These special classes of semigroups are important because they help us understand different ways that things can be combined together. Just like how we have different rules for adding, subtracting, multiplying and dividing numbers, these special semigroups have their own unique way of combining things.