Is there a general name for $(a\star b)$, $\star$ being any arbitrary ( binary) operation?

Question

$\bullet$ 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?

Answer 1

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.

Answer 2

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).

Answer 3

$a*b$ is in fact a simple example of a "word". But generally it would be called the product of $a$ and $b$.