Untermonoid

A submonoid is a subset of a monoid that is itself a monoid under the same binary operation and with the same identity element. Formally, let (M, ) be a monoid with identity element e. A subset S of M is a submonoid of M if the following conditions hold: 1. The identity element e of M is an element of S. 2. For any two elements a and b in S, their product a b is also in S. This second condition means that S is closed under the monoid operation. The set S, along with the operation restricted to S, forms a monoid. For example, consider the monoid of natural numbers with addition, (N, +), where the identity element is 0. The set of even natural numbers, {0, 2, 4, ...}, forms a submonoid because it contains 0 and the sum of any two even numbers is an even number. Similarly, in the monoid of all functions from a set X to itself under composition, the set of all bijections from X to X forms a submonoid, as the composition of two bijections is a bijection, and the identity function is a bijection. The concept of a submonoid is a fundamental notion in abstract algebra, analogous to subgroups in group theory or subrings in ring theory, but specifically for monoids.