2.11 Appendix I: A Brief Glossary of Algebraic Structures

Any algebraic structure is defined by a set of elements S and by one or more operations. The nature of the operations involved determines which of the axioms below are satisfied.

Consider a first binary operation ?

1. Closure wrt ?: if a and b are in S then a ? b is also in S.

2. Associative law wrt ?: ( a ? b) ? c = a ? ( b ? c).

3. Identity element wrt ?: There is a unique element e such that a ? e = e ? a = a for any a, ( e is often referred to as the zero element).

4. Inverse element wrt ?: For every a in S there is an inverse ? a such that a ? ? a = ? a ? a = e.

5. Commutative law wrt ?: a ? b = b ? a.

Consider a second binary operation ? that always takes precedence over operation ?

1. Closure wrt ?: if a and b are in S then a ? b is also in S.

2. Associative law wrt ?: ( a ? b) ? c = a ? ( b ? c).