What is the Absorbing element under addition on the extended real numbers?

Question

An Absorbing element is defined as a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. It comes from generalizing the zero under multiplication. absorbing element wiki

An absorbing element is unique by its properties so my question is on the extended real number line under addition which do we consider the absorbing element $\infty$ or $-\infty$? or does this set not have an absorbing element under this operation?

Answer 1

An absorbing element is unique in a semigroup. The problem is that the extended real numbers do not form a semigroup under addition, as explicitly stated in the Wikipedia entry Extended real number line. The structure you have is that of a partial semigroup: $a + (b + c)$ and $(a + b) + c$ are either equal or both undefined. But in such a partial structure, you may have several absorbing elements and indeed, $+\infty$ and $-\infty$ are both absorbing elements.

Answer 2

If $+\infty$ is an absorbing element under addition in the extended real line, $\overline{\mathbb{R}}$, then adding $+\infty$ to any element in $\overline{\mathbb{R}}$ should give $+\infty$. But $-\infty\in\overline{\mathbb{R}}$ and $(+\infty)+(-\infty)\neq(+\infty)$.