Function mapping notation for a binary operation on a set

Question

I found the following definition of a binary operation on a set from here:

A binary operation $*$ on a set $S$ is a map $\ast:S\times S \rightarrow S$

My question is, if I define an operation $\ast$, then should it be possible to reach every point in $S$ with some input in $S\times S$? What if I define an operation on a set for which the image was always in $S$, but could never be some parts of $S$?

The notation seems to suggest that this would not classify as a binary operation.

score 3 · Accepted Answer · answered Jul 01 '15 at 13:50

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Defining such an operation would be fine. It just depends on how useful you want your binary operation to be. A binary operation that reaches every element of the set is an essential part of a mathematical group, which is a very important area of study. But there are objects called semi-groups that may not have this property, and are still useful.

answered Jul 01 '15 at 13:50

Plutoro

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Great thank you, but what I wanted to ask was would the notation $\ast:S\times S \rightarrow S$ still be valid for such an operation? – texasflood Jul 01 '15 at 13:59
Yes, the function need not be surjective to have that notation. – Plutoro Jul 01 '15 at 14:01

score 2 · Answer 2 · answered Jul 01 '15 at 13:54

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You can certainly have binary operations on a set where not every element of the set is an output. For instance, addition on the positive integers is a lovely binary operation on the positive integers, but you can't reach $1$ as an output.

answered Jul 01 '15 at 13:54

paw88789

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Function mapping notation for a binary operation on a set

2 Answers2