Study Notes

Algebraic Structures: Definition, Types & Examples

An algebraic structure refers to a set endowed with structured operations that follow certain fundamental rules or axioms. These structures play a significant role in modern algebra and are used extensively across various branches of mathematics. They allow us to perform operations on the elements of these sets and transform them systematically.

Definition & Types

An algebraic structure consists of a nonempty set S endowed with a binary operation *. To qualify as an algebraic structure, S must satisfy certain properties or axioms. These include closure, associativity, and existence of an identity element. For example, given two real numbers x and y, if we have a group operation * that combines them to form another real number z (z = x * y), then {x, y} becomes an algebraic structure under this operation.

There are several types of algebraic structures, including groups, semi-groups, monoids, rings, fields, and vector spaces. Each type has specific axioms and properties associated with their respective operations.

Hands-On Problem

Consider the following set S = {1, -1}. Prove that S is an algebraic structure under addition (+).

Solution

To prove that S is an algebraic structure under addition (+), we need to show that it satisfies the following properties:

Closure: (a + b) belongs to S for all a, b in S. In our case, 1 + (-1) = 0 and -1 + 1 = 0, both of which belong to S. Therefore, S satisfies the closure property for addition.
Associativity: a + (b + c) = (a + b) + c for all a, b, c in S. Since 1 + ((1 + 1) - 1) = (1 + 1) + 1 = 1 and -1 + ((-1 + 1) - 1) = (-1 + 1) + 1 = 1, we have associativity for addition in S.

Conclusion

In conclusion, algebraic structures are essential in modern algebra, providing a foundation for understanding the relationships between sets and their elements through structured operations. Various types of algebraic structures have distinct properties and applications in diverse areas of mathematics.