Semialgebraic group

Definition over the real numbers
A semialgebraic group over the reals, also called a real semialgebraic group, is a group object in the category of real semialgebraic varieties. Here, a real semialgebraic variety is something that can be obtained by iterating finitely many union and intersection operations starting either with the zero sets of polynomials or the sets of points where a polynomial is positive.

Definition over fields similar to the reals
The same definition as for the field of real numbers works over any real-closed field, with the theory remaining the same.

The definition can also be interpreted over any ordered field. However, not all aspects of the theory remain the same once we shift to arbitrary ordered fields.