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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?

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  • $\begingroup$ There is no real number $x$ such that $x+y=x \;\forall y$. Multiplication, however, admits $0$ as an absorbing element as $0\times y=0\;\forall y$. $\endgroup$
    – lulu
    Commented Oct 13, 2016 at 16:55
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    $\begingroup$ thats why i said extended reals it includes infinities. @lulu $\endgroup$
    – shai horowitz
    Commented Oct 13, 2016 at 16:55
  • $\begingroup$ ok, but what is your definition? Are you including two points, $+\infty$ and $-\infty$? If so, what do they add to? If you just add one point and define $\infty+y=\infty\;\forall y\in \mathbb R\cup \infty$ then, sure. $\endgroup$
    – lulu
    Commented Oct 13, 2016 at 16:57
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    $\begingroup$ Just to say, if you are following the conventions of, say, this definition then $\infty +(-\infty)$ is not defined, so you haven't got a binary operation. $\endgroup$
    – lulu
    Commented Oct 13, 2016 at 17:01
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    $\begingroup$ What if we projectively extend the Real numbers, i.e. unify positive and negative infinity as a single value. en.wikipedia.org/wiki/Projectively_extended_real_line $\endgroup$
    – shai horowitz
    Commented Oct 13, 2016 at 17:08

2 Answers 2

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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.

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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 $(+\infty)+(-\infty)\neq(+\infty)$.

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