In Algebra , semigroup is an algebraic system determined by the pair <G, *> , such that G is a nonempty set and * is a law of internal composition; therefore it fulfills the closing property, and is also associative.
In the case that <G, *> is a semigroup and the operation is commutative, it is said to be a commutative or abelian semigroup .
Summary
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 3 Other cases
 4 Affirmations
 5 References
 6
Definition.
Let be a set G and the binary operation * defined as * (x, y) = z , normally written as x * y = z that satisfies the axioms :
 Closing: . * is closed.
 Associativity: For all x , y , z of G , (x * y) * z = x * (y * z) .
It is said that G with the operation * is a semigroup .
Note that semigroups differ from groups in that these are semigroups that also have a neutral element and for each element in the set they have one and only one inverse element .
Examples.
 Every group <G, *>is a semigroup.
 The following are semigroups represented in tabular form:
<{a, b}, @>  <{a, b, c}, *>  


Other cases
 Let S be a nonempty set to a fixed element, let us define x * y = a, for any x, and elements of S. The closure of * is true.
The associative property: (x * y) * z = a * z = a = x * a = x * (z * t). Trivially, every set not empty with the law of internal composition *, is a semigroup. ^{[one]}
 The set N = {0,1,2 … n, …} of the naturals with addition and multiplication is a semigroup, since these two operations are laws of internal composition, that is, each pair of n . natural corresponds to a natural number that is its sum or product depending on the case. However, N is not a semigroup with the subtraction, since this is a partially defined operation, since for some ordered pairs of n. natural there is no difference; the same for the division of n. natural.
 In the case of integers, subtraction is a fully defined operation, it is a law of internal composition; however, the associative property is not fulfilled; for cases a (bc) and (ab) c in almost all cases. So the set Z of n. integers is not a semigroup with subtraction.
This same set of n. integers is a semigroup with the multiplication of even integers; since the product of two even numbers is equal and the associative property is fulfilled.
 Let T be the set of the powers of three, being T = {3 ^{n}: n is an integer} the set T with the multiplication is a semigroup, since the product of two powers of 3 is a power of 3; In addition, the product in T inherits the associative property of the product of the integers, since it is a subset of Z.
 The set of odd integers with multiplication is a semigroup, since the product of odd is odd, it is also associative; but it is not semigroup with the addition, since the sum of two odd numbers is even, although the associativity is fulfilled.
 The set of prime integers is semigroup with neither addition nor multiplication; for the sum of two prime integers, in almost all cases, is not prime; and in the case of the product of two primes, the primality is broken, the product already has four divisors and therefore it is no longer prime.
Affirmations
 Let t = {… [(b _{1}b _{2} ) b _{3} ] a _{4} …} a product of n elements b _{1} , b _{2} , … b _{n} of a multiplicative semigroup, then any product t ‘formed by the factors b _{1} , b _{2} , … b _{n} taken in that order is equal to t. This is the case of generalized associative property.
 In a semigroup it is possible to speak of the kth powerof an element b , product of k elements b . It is denoted b ^{k} , carrying the rule
b ^{i} b ^{j} = b ^{i + j} .
 In the case of an abeliancommutative semigroup, the product
t = b _{1} b _{2} … b _{m} does not depend on the order of the factors.
 Every homomorphic image of a semigroup is a semigroup.
 In a semigroup S, center C is a stable (occasionally empty) part