Semigroup

A semigroup is a special kind of mathematical structure. It has a group of items, called a set, and a way to combine any two items from that set. This combining method needs to follow a rule called associativity, which means the way you group the combinations does not change the result.

One easy example of a semigroup is joining together pieces of text, called strings. If you have the words "spot" and "run," putting them together gives "spot run." Associativity here means that joining "See" to the already joined "spot run" gives the same result as joining "See spot" to "run" — both make "See spot run."

People started studying semigroups carefully in the early 1900s. They found many interesting patterns and ways to understand them. Today, semigroups are important in computer science, especially in how machines follow sets of rules, and they also help in studying chance processes and solving certain kinds of equations.

Algebraic overview

A semigroup is a special kind of math structure. It has a set of things and a way to combine any two of them, called an operation. This operation must follow a rule called "associativity," which means the way you group the combinations doesn’t change the result.

Semigroups are like a simpler version of groups, but they don’t need an “identity element” (a special item that doesn’t change other items when combined with them) or “inverses” (items that can undo each other). For example, joining strings together is a semigroup operation, but only if you don’t include an empty string as the identity. Positive whole numbers with addition are another example of a semigroup.

Definition

A semigroup is a collection of items, called a set, along with a special way to combine any two items from the set. This combining method must follow a rule called the associative property. This means that when we combine three items, it doesn't matter which pair we combine first — the result will be the same.

In simpler terms, a semigroup is just a set with a way to combine its items that works consistently every time.

Examples of semigroups

Here are some simple examples of semigroups:

• The empty set can be a semigroup with a special empty operation.
• A semigroup can have just one element, where that element combined with itself gives the same element.
• There are five different ways to make a semigroup with two elements.
• You can take any set of numbers and use addition to combine them, and this creates a semigroup.
• The set of all strings of letters, where you join strings together to make new strings, is also a semigroup.

These examples show how different kinds of sets and operations can form semigroups in mathematics.

Basic concepts

In mathematics, a semigroup is a special kind of structure. It has a set of elements and a way to combine any two elements, called an operation. This operation must follow a rule called "associativity," which means the way you group the operations does not change the result.

Semigroups can have special elements, like identities, which act similarly to the number 1 in multiplication. For example, if you combine an identity element with any other element, the result is that other element. Some semigroups also have "zero" elements, which when combined with any other element, give that zero element as the result.

Structure of semigroups

In a semigroup, any smaller part of the set can create a smaller semigroup that includes it. We say this smaller part generates that semigroup. One single element can create a subsemigroup by repeating itself many times. If this repeats only a few times, the element is of finite order; otherwise, it is of infinite order. A semigroup where every element repeats is called periodic.

A subsemigroup that also works like a group is called a subgroup. Subgroups and special elements called idempotents are closely linked. Each subgroup has exactly one idempotent, which is its identity element. For each idempotent, there is a largest subgroup that includes it, and every large subgroup comes from an idempotent this way.

When the semigroup has a limited number of elements, we can learn more. Every such semigroup is periodic and has a smallest part called an ideal and at least one idempotent. There are more semigroups of a certain size than there are groups of that size. For example, for a set with two elements, eight different ways work as semigroups, but only two work as groups. For more details on finite semigroups, see Krohn–Rhodes theory.

Special classes of semigroups

Main article: Special classes of semigroups

A monoid is a type of semigroup that has a special element called an identity element. A group is a monoid where every element has a matching inverse element. A subsemigroup is a smaller group taken from a bigger semigroup, where the smaller group still follows the same rules.

Some other special types of semigroups include cancellative semigroups, where certain rules help us figure out answers, and bands, where doing the operation twice gives the same result as doing it once. There are also semilattices, which follow extra rules, and transformation semigroups, which help us understand how things change in steps. These ideas are useful in studying machines and systems that follow steps.

Structure theorem for commutative semigroups

A commutative semigroup can be understood using special sets called semilattices. A semilattice is a set where every two elements have a greatest lower bound, and this lower bound operation turns the semilattice into a semigroup.

When we connect a semigroup to a semilattice using a special mapping, each group of elements in the semigroup becomes its own small semigroup. These small groups are organized by the semilattice in a specific way.

The structure theorem explains that any commutative semigroup can be broken down in a special manner so that its simplest form becomes a semilattice. Each piece of this broken-down semigroup has a special property called the Archimedean property, which helps describe how elements relate to each other.

Group of fractions

The group of fractions of a semigroup is a way to turn it into a group. We start with a semigroup S and create a group G that includes all elements of S. This group follows all the rules that the semigroup already has. There is a special mapping from S to G that keeps these rules true.

One important question is when this mapping works perfectly, meaning each element of S stays unique in G. This is not always true. For example, if S is made from sets using intersection, the group ends up very small. For some semigroups, special rules must be met for this mapping to work well. When these rules are met, we can build the group in a clear way.

Semigroup methods in partial differential equations

Further information: C0-semigroup

Semigroup theory helps us understand some problems in partial differential equations. Think of it like this: we can treat a problem that changes over time as if it were a regular equation, but instead of working with numbers, we work with functions.

For example, let's look at the heat equation, which describes how heat spreads out over time. We can rewrite this problem using semigroups. This lets us use tools from a different area of math to find solutions. The key idea is that the solution can be thought of as a special kind of math operation applied to the starting condition. This operation is called a semigroup, and it helps us move from the beginning state to the state at any later time.

History

People started studying semigroups later than other math ideas like groups or rings. One person first used the word for semigroups in French in 1904, and then others used it in English in 1908.

A mathematician named Anton Sushkevich did important early work on semigroups in 1928. After that, many other mathematicians helped build the foundations of semigroup theory. Two of them even wrote big books about it in the 1960s. In 1970, a special magazine just for semigroup theory started.

Later, more focus went into special types of semigroups and how they can be used in other areas of math.

Generalizations

If we stop needing the rule called "associativity" for a semigroup, we get something called a magma. This is just a group with a binary operation, but without the special rule.

We can also think about semigroups with more than two pieces at once. These are called n-ary semigroups. For example, a ternary semigroup uses three pieces together, and has its own special rule. A 2-ary semigroup is just a normal semigroup.

There is also something called a semigroupoid, which is like a category but does not need every pair to have an identity.

Sometimes people also think about versions of semigroups that use infinitely many pieces at once.