# Clarifications about the definition of algebraic systems and algebraic structures

I am learning about Groups from a discrete mathematics textbook for a computer science course by Grimaldi.

An algebraic system is defined as a set along with operations on elements of the set. An algebraic structure is defined as a set, operations on elements of the set and relations between elements of the set which leads to a structure on the elements of a set.

The author then goes on to talk about groups as being algebraic structures(so does wikipedia). I am failing to see what structure a group imposes on a set it is defined on. This is making me think about a group an algebraic system instead of an algebraic structure.

What should I see in a group that will help me relate it the definition of an algebraic structure? What should I understand by "structure on the elements of a set"? Explanation with examples would help a lot.

EDIT - Exact definitions as given in the textbook
An algebraic system is a system consisting of a nonempty set $A$ and one or more n-ary operations on the set $A$. It is denoted by $\langle A, f_1, f_2, ... \rangle$.
An algebraic structure is an algebraic system, $\langle A, f_1, f_2, ... , R_1, R_2, ...\rangle$, wherein addition to operations $f_i$, the relations $R_i$ are defined on A. This leads to a structure on the elements of A.

• Your definitions are rather vague. Can you give exact quotations? – Zhen Lin Dec 19 '12 at 5:13
• @ZhenLin, I've added the exact definitions. – Abhijith Madhav Dec 19 '12 at 5:45
• Hmmm. I think by ‘algebraic structure’ most people mean a set with operations but no relations. – Zhen Lin Dec 19 '12 at 7:33
• Following the definitions, an algebraic system is for example $\langle \Bbb Z, +\rangle$ and an algebraic structure is $\langle \Bbb Z,+,\leq\rangle$ – leo Dec 19 '12 at 20:34

Maybe the easiest example is the integers. As a set, $\mathbb{Z}$ is just a collection of elements which have nothing to do with each other - $0$ is no more special than $79$, $10$ and $-10$ have no particular relationship. But once we consider the integers as a group under the binary operation 'addition' (which I think we're all familiar with) we are able to see that $10+(-10)=0$ and $z + 0 = z$ for every $z\in \mathbb{Z}$.
• You probably mean addition modulo $n$ in your definition of $\mathbb Z_n$. Also, all groups can be realized as groups of permutations, not just finite groups. – Santiago Canez Dec 19 '12 at 5:01