41. A topological group is a mathematical object with both an algebraic structure and a topological structure. 42. This proof does not work for non-algebraic structures . 43. Important other algebraic structures such as rings arise when requiring only part of the above axioms. 44. The idea of describing algebraic structures with finite-automata can be generalized from groups to other structures. 45. A forgetful functor between categories of algebraic structures " forgets " a part of a structure. 46. A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures . 47. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures , such as fields. 48. But over the years he became known for his work in applying algebraic structures to real-life situations. 49. Early results include ordered algebraic structures . 50. Suppose A is an algebraic structure .
