Is there a general name for $(a\star b)$, $\star$ being any arbitrary ( binary) operation? $\bullet$ We have a name for the image of $( a, b)$ under the operation of addition, namely a sum. 
$\bullet$ We have a name for the image of $( a, b)$ under the operation of multiplication , namely a product. 
$\bullet$ We have other names such as quotient, or difference for other operations. 
It seems that it would be useful to have a name denoting in general the image of an arbitrary ordered pair $(a,b)$ under an arbitrary operation $\star$. 
Is there such a name? 
 A: A binary operator can be construed as a binary function we just like to write without the usual $f(a,\,b)$ notation. Then $a\star b$ is an image of the operator/function.
As others have noted, in certain contexts product is also used. (Presumably, it won out over sum because "addition" is usually reserved for a commutative operation, with one obvious exception).
A: To the best of my knowledge, there is no universally understood term which will denote the result of applying a binary operation to two objects.  As such, if you decide to use such a term, you should be careful to define that term before you start using it willy-nilly.  That being said, there seem to be several options, include those mentioned in the comments.  I am presenting them is the order that, in my opinion, runs from worst to best (that is, the best options are at the bottom).


*

*Sum: The object $a \star b$ is the sum of $a$ and $b$.  In general, I would say that calling it a sum implies that $\star$ is a commutative operation.  If it is not commutative, then "sum" may be inappropriate.

*Composite: The object $a\star b$ is the composition or composite of $a$ and $b$.  In my experience, function composition is often the "multiplicative operation" of an algebra.  For example, multiplication of two $n\times n$ matrices can be seen as the composition of the two linear transformations represented by those matrices—this notion generalizes nicely to spaces of linear functionals (for example).  Personally, I would think that "composition" implies noncommutativity, but I can't back that up with anything other than my own gut.

*Product: The object $a \star b$ is the product of $a$ and $b$.  This is likely to be an easily understood term—an arbitrary binary operation can often be understood as a generalized version of addition or multiplication.  As noted above, "sum" implies commutativity, while "product" does not (though multiplication can be commutative, so there is no loss of generality).

*Result:  The object $a\star b$ is the result of starring $a$ and $b$.  That is, come up with a noun to describe the operation (this is the "star operation"), verb that noun, then use the phrase suggested above.

*Image: See J.G.'s answer.

*$\star$-product:  Finally, saving the best for last, call $a\star b$ the $\star$-product of $a$ and $b$.  Read aloud, if $c = a\star b$, then say

$c$ is the star-product of $a$ and $b$.

I think that this is likely to be completely unambiguous, and offers the most bang for the buck.
A: $a*b$ is in fact a simple example of a "word".  But generally it would be called the product of $a$ and $b$.
