Is there a name for an element $z$ such that $\,zx = z = xz\,$ for all $x$?

Question

Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc):

1. For any given set, the intersection with the empty set results in the empty set.
2. For any real number, multiplication with zero results in zero.

Answer 1

I've heard absorbing element or annihilating element used for this (Wikipedia link).

So in your second example, we'd say that $0$ is an absorbing element for the operation $\times$ on $\mathbb{R}$.

In your first example, you'd have to specify a set in which you're working. I'd suggest formulating this as: for any set $S$, the element $\varnothing\in\mathcal{P}(S)$ is an absorbing element for the operation $\cap\,$.

Answer 2

(2) is a consequence of the ring axioms (see https://en.wikipedia.org/wiki/Ring_(mathematics)); in general, an element $x$ with the property that $x*y=x$ for every $y$ is called an annihilator for the operation $*$.

For (1), the set of subsets of $X$ (for $X$ fixed) forms a Boolean algebra (see https://en.wikipedia.org/wiki/Boolean_algebra; this is the structure provided by intersection, union, and complement). Boolean algebras always have a least element $l$, with the property that $l\wedge x=l$ for every $x$ (where "$\wedge$" is intersection, in the case mentioned above).

Answer 3

I would like to propose some definitions. All deal with elements in a monoid $M$.

Proposal:

Let the absorbing element or the absorbor (under multiplication) refer to the unique element $x$ such that for all $y$ in $M$, $xy = x$. This is also called the zero element or zero, denoted $0$. This element does not necessarily exist in $M$.

Let an absorbing element or an absorbor (under multiplication) refer to any element $x$ such that there exists some $y \neq 1$ such that $xy=x$.

Let the annihilating element or the annihilator (under multiplication) refer to the unique element $x$ such that for all $y$ in $M$, $xy = 0$. This element, if it exists, turns out to be zero.

Let an annihilating element or an annihilator (under multiplication) refer to any element $x$ such that there exists some $y \neq 0$ such that $xy=0$.

Defense:

I think there are places where we can absorb an element, but not send it to $0$. If I introduce a new operation, I can have an absorbing element $a$ under that new operation where $ab \neq 0$. This is the reason for the distinction between absorption and annihilation.

Also, if $M$ is a module over a ring $R$, and $m \in M$ and $r \in R$ are both nonzero, then I could have $rm = 0$. In which case, I would say that $r$ is "an annihilator" even though it is not "the annihilator". This is the reason for defining the new notions that use "an" and are not unique. Similarly, we can say that $m$ is "an annihilatee" because it has an annihilator. The set of all annihilatees in $M$ is called the torsion of $M$.

Thanks, would love to hear your opinions on this.