hemiryhmät

Hemiryhmät, often translated as "hemigroup" or "half-group," is a mathematical structure that shares some properties with groups but lacks others. In abstract algebra, a hemigroup is a set equipped with a single associative binary operation. This means that for any elements a, b, and c in the set, the operation satisfies (a b) c = a (b c).

The key difference between a hemigroup and a group is the requirement for inverses. In a group, every element must have an inverse such that when an element is combined with its inverse, the result is the identity element. Hemigroups do not have this requirement. Consequently, there might not be an identity element, and even if there is, not every element necessarily possesses an inverse within the hemigroup.

Examples of hemigroups can be found in various areas of mathematics. The set of all n x n matrices with entries from a field, under matrix multiplication, forms a hemigroup. If the matrices are invertible, they form a group, but the set of all matrices includes non-invertible ones. Another example is the set of natural numbers with addition, where the operation is associative, but there is no identity element (0 is sometimes included, but not always considered part of the natural numbers in all contexts) and no additive inverses for positive integers.

The study of hemigroups is a branch of semigroup theory, which deals with sets and associative binary operations. While semigroups encompass all hemigroups, hemigroups represent a more general case where the existence of an identity element and inverses is not guaranteed. Their properties and classifications are subjects of ongoing mathematical research.