A (sub)semigroup with an endomorphism is a mathematical structure that consists of a set of elements and a binary operation, along with an endomorphism (a function that maps an element to itself) defined on the set. This structure is closed under the binary operation and the endomorphism, meaning that the result of applying the operation or endomorphism to any two elements in the set will always produce another element in the set.
A regular semigroup is a mathematical structure that also consists of a set of elements and a binary operation, but does not have an endomorphism defined on it. In other words, a regular semigroup does not have a function that maps an element to itself. This means that a (sub)semigroup with an endomorphism has an additional layer of structure and symmetry compared to a regular semigroup.
One example is the set of all real numbers under addition, with the endomorphism being squaring the numbers. Another example is the set of all positive integers under multiplication, with the endomorphism being taking the reciprocal of the numbers. In general, any algebraic structure that has a binary operation and an endomorphism defined on it can be considered a (sub)semigroup with an endomorphism.
(Sub)semigroups with endomorphisms have applications in various fields such as algebra, number theory, and computer science. They can be used to study properties of more complex algebraic structures, and their study can lead to insights and advancements in these fields.
(Sub)semigroups with endomorphisms have closure, associativity, and identity properties, just like regular semigroups. However, they can also have additional properties such as commutativity, inverse elements, and idempotence, depending on the specific endomorphism defined on the structure. They can also be classified into different types based on the endomorphism used, such as idempotent, nilpotent, and group-like (sub)semigroups.