Semigroup is a basic concept in algebra and one of the transcendental knowledge of group theory. Semigroup is a mathematical structure, which consists of a non-empty set and binary operations on this set. The operations in a semigroup must satisfy the closure and association laws, but they do not need to meet the requirements of other groups.
We define a set S, which contains elements A, B, C and so on, and also contains a binary operation *. If this operation satisfies the associative law for any a, b and c∈S, that is, (a*b)*c=a*(b*c).
Then we say that this set S and this operation * form a semigroup. Where a, b and c are called elements of semigroup S, and operation * is called closed binary operation of this semigroup.
It is worth noting that in semigroups, operations are not required to have identity elements or inverse elements. This is because the main function of semigroups is to explore the properties of mathematical structures, not to apply them to problems. Therefore, the theoretical study of semigroups is often extended to other mathematical fields, such as communication theory and computer science.
Semigroups are different from groups. They must satisfy three axioms: closeness, associative law and the existence of inverse elements, and they also need to satisfy the existence of a unit element. Therefore, it can be said that semigroups are special cases of groups and are the basis of studying groups.
Semigroups are widely used in mathematics, physics, chemistry, computer science and other fields because they are easy to understand.
For example, in computer science, semigroups are widely used in computer programming and theoretical research, which can describe the program complexity under a specific computing model; In physics, the application of semigroups in the research of quantum mechanics and relativity can provide more accurate mathematical models.
In a word, semigroup is one of the basic concepts in algebra. It consists of a set and an operation, which is easy to understand. Semigroups are different from groups, but they can provide an important basis for the study of groups. Semigroups are widely used in mathematics, physics, computer science and other fields, and also have potential applications in people's daily life.