During studying the text book of abstract algebra by john Farleigh, I encounter with tye definition of binary Algebraic structure. Then I tried to find the difference between binary Algebraic structure and Algebraic structure. I checked on many resources like Wikipedia, proofWiki, and MSE but instead of understanding notion I got confused.
Now I am confused in the definition of
1- Algebraic structure
2- binary Algebraic structure
3- Algebraic system
4- Algebraic operation.
5- Magma.
In the book author says:
A binary Algebraic structure $ (S,*) $ to be a set S together with a binary operation $ * $ on it.
But after searching alot, I think the definitions are :
Algebraic structure:
An Algebraic structure is a non empty set $S$ together with n binary operations, say $ *_1, *_2, \cdots , *_n $ on
$S$. It is denoted as $( S, *_1, \cdots , *_n )$ or sometimes as $\langle S, *_1, \cdots , *_n \rangle $
Binary Algebraic structure:
An Algebraic structure consisting of only one binary operation.
Algebraic system:
An Algebraic system is the collection of non empty set $S$ together with set $O$ of finitary operations on $S$, it is denoted by $(S,O)$.
Algebraic operation:
An Algebraic operation $ * $ on a non empty set $S$ is a function $$ *:A^n → A$$
Magma:
A binary Algebraic structure is also called magma or groupoid.
Are these all definitions correct? I have specially doubt on Algebraic structure and Algebraic system.
Thank you.
