What is the correct notation to denote the ordered/unordered concatenation of two strings?

I'm am developing some notation for a model and am needing to denote the concatenation of two strings. However, I'm not sure if there is a standard way to do this.

Let's take a simple example. Say I have variables $X$ and $Y$ which take the values of $a$ or $b$ (in the context I'm working with, $a$ and $b$ are strings with a specific value) and $Z$ which is a concatenation of $X$ and $Y$. Now, I want to define concatenation, $X + Y = Z$, such that,

$a + a = aa$

$a + b = b + a = ab$

$b + b = bb$

I have two questions here.

1) I don't want to use the "+" symbol to denote concatenation and was wondering if anyone knows if there is a standard mathematical notation for this.

2) I am also unsure whether one calls this unordered concatenation (as $a+b = b+a$) or ordered concatenation (as $a$ is to always come before $b$ in the concatenated string $Z$).

• Pick whatever notation you want. – Mariano Suárez-Álvarez Jan 31 '18 at 21:04
• As for the rest of what you wrote, I am afraid I was not able to understand at all what you want.ç – Mariano Suárez-Álvarez Jan 31 '18 at 21:05
• Besides issues of notation, what is the definition of "unordered concatenation"? I cannot imagine how one would alter the concept of concatenation of strings so as to obtain a commutative operation. – Lee Mosher Jan 31 '18 at 21:12

A common notation for strings is to use the pipe symbol $|$. This is often used in cryptography and coding theory. In general just decide on the convention you wish to use and indicate it at the top of the document you are writing.