4 elements set and a binary operation

Question

As part of a larger problem I am working on, I need to come up with an example as follows: A set of exactly 4 elements with a binary operation. The set plus the operation have to comply with all the requirements of a group with the exception of the last one. I.e. Invertibility. It can be a known operation or a made up one * with a table. It is important though that invertibilty will not exist for a least one of the elements. In other words the 4 elements set, exhibits the first 3 qualities of a group but not the last one.

I tried thinking of a known operation such as addition but could not find an example that will maintain closure. With a made up operation and a table it is very difficult to check for associativity so I'm lost there as well.

Any help would be appreciated. Cheers

Answer 1

Consider a set $A=\{1,2\}$ with two elements, now consider it's power set $$P(A)=\{\emptyset, \{1\}, \{2\}, \{1,2\}\}.$$ with four elements and the binary operation as intersection of two sets. $(P(A), \cap)$ is a monoid (group without requiring existence of inverses) with identity element as $A$ but inverse need not exist.

Answer 2

Ordinary multiplication, over the set $\left\{0,1,\frac{-1+\sqrt{-3}}{2},\frac{-1-\sqrt{-3}}{2}\right\}$.

Answer 3

A fairly obvious example is $Z_n=\{0,1,\ldots,n-1\}$ under the operation of multiplication modulo $n$. (Here $n=4$.)

My other example (now deleted) was $\{0,1,\ldots,n\}$ under the operation $a*b=\min(a+b,n)$. (Here $n=3$.)