Examples of Algebraic Structures Not Forming Groups

Algebraic structures are fundamental concepts in modern mathematics, playing a crucial role in various fields including abstract algebra, computer science, and theoretical physics. While groups are one of the most well-known and studied algebraic structures, there are many algebraic structures that do not form groups. This article explores a common example of such structures, the semigroup, as well as other examples and their implications.

What is a Semigroup?

A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation but not necessarily having an identity element or inverses for its elements. This concept is important in understanding the hierarchy and nuances within algebraic structures.

Example of a Semigroup: Positive Integers with Addition

Definition and Properties:

Closure: For any two positive integers (a) and (b), (a b) is also a positive integer. Associativity: Addition is associative, meaning (a (b c) (a b) c) for all (a, b, c in mathbb{Z}^ ). Identity Element: There is no identity element within (mathbb{Z}^ ) for addition because there is no positive integer (e) such that (a e a) for all (a in mathbb{Z}^ ).

Despite satisfying closure and associativity, the set of positive integers (mathbb{Z}^ ) with addition does not form a group. This is due to the lack of an identity element, making it a semigroup but not a group.

Is Multiplication on the Integers a Group?

No, multiplication on the integers does not form a group because not all elements have multiplicative inverses. As an algebraic structure, multiplication on integers satisfies closure and associativity, but it fails to satisfy the requirement of having an identity element for all elements. Specifically, except for the number 1, no other integer has a multiplicative inverse within the set of integers.

Other Examples of Non-Group Algebraic Structures

Rings and Fields

Another important class of algebraic structures that can be non-groups are rings and fields. In a ring, the set is equipped with two binary operations (addition and multiplication), with both operations being associative, the addition having an identity element, and the multiplication being distributive over addition. A field is a special kind of ring where the non-zero elements have multiplicative inverses.

For example, the set of integers (mathbb{Z}) under addition and multiplication forms a ring but not a field, as not all non-zero integers have multiplicative inverses. However, the set of rational numbers (mathbb{Q}) forms a field.

Monoids

A monoid is an algebraic structure similar to a group but with a different requirement. A monoid consists of a set equipped with an associative binary operation and an identity element, but not necessarily with inverses for all elements. Like a semigroup, a monoid can be thought of as a generalization of a group without the requirement for inverses.

Hopf Algebras, Division Algebras, and Others

Beyond semigroups and monoids, there are even more complex algebraic structures such as Hopf algebras, division algebras, and lambda rings which can also fail to be groups. These structures often have additional properties and operations, and while they may not form groups, they still play significant roles in advanced mathematics.

Conclusion

The study of algebraic structures not forming groups, such as semigroups, rings, and fields, provides valuable insights into the nature of mathematical operations and their properties. Understanding these structures and their limitations helps mathematicians and researchers to develop more robust theories and applications in various fields.