# Is there a formal term for the placement of two symbols next to each other to imply an operation?

An example of what I am talking about is indicating multiplication by writing $$ab\equiv{a}\times{b},$$ in traditional real number algebra.

I was writing some notes involving matrix multiplication. Previously in these notes I had specified that placing two tensor symbols next to each other indicates a direct product. For example:

Let $\mathfrak{A}=\left\{A_{ij}\right\}_{n\times{n}}$ and $\mathfrak{v}=\left\{v_{k}\right\}_{n\times{1}}.$ So in the context of my definition:

$$\mathfrak{A}\mathfrak{v}\equiv{\mathfrak{A}\otimes\mathfrak{v}}\equiv{\left\{A_{ij}v_{k}\right\}_{n\times{n}\times{n}}}.$$

But when working with matrices in linear algebra it is common practice to use

$$\mathfrak{A}\mathfrak{v}\equiv{\left\{\sum_k A_{ik}v_{k}\right\}_{n\times{1}}}.$$

Well, I want to use the latter definition in one example. When I tried to state that placing the symbols next to each other without any operator symbol between them means matrix multiplication in this example, I found that I have no formal terminology for doing that. I started to say "the juxtaposition of symbols...", but when I looked up the work juxtapose, I realized it's not what I mean. It connotes an intent to compare and contrast.

Is there a formal term for implying an operation by placing two symbols next to each other?

In abstract algebra, when words are formed using letters from an alphabet, the operation of joining two words together with no symbol in-between is called concatenation.

Now I wouldn't recommend that in a case where the left and right sides are mathematical objects of a different nature. In that case, it seems that juxtaposition is fine - I have seen it used at least a couple of times, and I don't feel that it carries a connotation in a mathematical context. And if you want to be extra clear, you can add some words the first time you use it, in a footnote for instance.

• while I am not a matematician, I work a bit with inverse problems. Its pretty common to describe linear problems as $Ax=b$, in fact I have been called out for using symbols to represent the multiplication of $A$ and $x$ when "they are not needed". – Ander Biguri Aug 6 '18 at 10:29
• @AnderBiguri absolutely, there are a number of situations where it is commonplace to not actually write down operators, and where doing so is even a little bit awkward. The only situation where I'd actually write "$\times$" or "$\cdot$" is when a confusion may arise otherwise, as in $2\times 3$ ;-) – Arnaud Mortier Aug 6 '18 at 10:38
• Oh, and I always though it was clear to write $23$! :) – Ander Biguri Aug 6 '18 at 10:40

You are correct about juxtaposition. The Wikipedia article Order of operations has this sentence:

However, in some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division

which indicates that this juxtaposition is used in the context of precedence of operators in mathematical expressions. It is all in the associated context and defined by conventions. You can use, modify, or create your own conventions as long as you explain what you are doing.

The third term I have heard for this is apposition, as in "two symbols written in apposition".

(However, I concur with the other answers that juxtaposition is perfectly understandable.)

I would write it as: "In this paper, direct product is used as the implicit operator for tensors ($\mathfrak{A}\mathfrak{v}\equiv{\mathfrak{A}\otimes\mathfrak{v}}$), while matrix multiplication is retained as the implicit operator for matrices and vectors."

However, using concatenation with unusual symbols carries risks. When I first saw $\mathfrak{A}\mathfrak{v}$, I thought it was a script W. It could also be interpreted as 21v in a strange script. Perhaps your audience is more familiar with this notation, but if you want your paper to be accessible to a more general audience, you should consider other choices for notation, such as explicitly writing the direct product operator, using symbols other than letters in Fraktur, or at least putting a space between them.