Algebra over a set refers to a collection of mathematical operations that utilize a set of numbers or elements to perform various calculations. It's an essential concept in abstract algebra that has a multitude of synonyms. For instance, algebra over a set can also be referred to as abstract algebra, universal algebra, or mathematical algebra. Additionally, it's frequently called a ring theory, ring algebra, associative algebra, or algebra with an identity element. Other people may use the term commutative algebra or noncommutative algebra to refer to algebra over a set. Overall, algebra over a set is a crucial component of mathematics, and it's essential to understand its various synonyms to communicate effectively with other mathematicians.