What does $*$ mean in equivalence relations?

Question

The notation of "$*$" started being used in my proof textbook in the section of equivalence relations and partitions yet it never once said what it means.

An example from the textbook:

Let $\mathbb{Z}^* = \mathbb{Z} - \{0\}.$ Define the relation on $\mathbb{Z} \times \mathbb{Z^*}$ by, for all $a,c \in \mathbb{Z}$ and all $b,d \in \mathbb{Z}$

What does "$*$" mean?

Answer 1

In an algebraic context, many authors use $A^*$ to denote the set $A$ without the zero element. In your specific example, the author uses $\mathbb Z^*$ to denote the set $\mathbb Z$ without $0$, i.e. $\mathbb Z-\{0\}$. So, $*$ is just used in a context of notation and does not denote any particular operation.

In a similar manner, it also common to write $\mathbb R^*$ for $\mathbb R-\{0\}$, $\mathbb C^*$ for $\mathbb C-\{0\}$, etc.

Answer 2

The notation refers to all non-zero elementes, in your case, of the integers $\mathbb Z$. As it is defined by excluding $0$ from the integers by $\mathbb Z^*=\mathbb Z-\{0\}=\mathbb Z\setminus\{0\}$.