Algebraic systems are mathematical structures that consist of a set of elements and operations defined on those elements. Common examples include groups, rings, and fields. Each system has specific properties that govern how the elements interact, such as closure, associativity, and the existence of identity and inverse elements. These systems are fundamental in abstract algebra and are used to study symmetry, number theory, and many other areas of mathematics. Understanding algebraic systems helps in solving equations and analyzing mathematical relationships, making them essential in both theoretical and applied mathematics.