Does there exist an infinite magma with the following property: Let $n$ be a positive integer greater than or equal to $2$. For all $x_1,...,x_n$, if $x_1,...x_n$ are all distinct, then all products where one and only one each of $x_1,...,x_n$ occurs are also distinct. Also, this property has to hold for all $n \geq 2$, like $2,3,4,5,$ etc. So, basically this property would say, for example, that for all $x$ and $y$, if $x \neq y$, then $(x*y) \neq (y*x)$, and for all distinct $x$, $y$, and $z$, the $12$ products that are possible where one and only one each of $x$, $y$, and $z$ occur are all distinct, and so on for $4$ variables, $5$ variables, etc.
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Yes, if I understand you correctly. If x and y are any two strings, define the operator x*y as the concatenated string •xy.
Then define the set G recursively by saying the strings 0 and 1 are members of G, and if x and y are any strings in G, then x*y is also a member of G.
Then G is a magma under the * operator (which is neither commutative nor associative). Note that all of the products of G are unique because you can tell how any string was formed just by reading it. For example, the string •x•xy was formed by x*(x*y). Hence it has the desired property.
It is called the free magma on 0 and 1. (You can look up more information about free magmas elsewhere.)
In another interpretation, your desired property is not possible for n≥3 because if a=x, b=y, and $c=x*y$ are distinct, then the products $c*(a*b) = c*c = (a*b)*c$ must be equal.
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