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In multiplication, x • 1 = x. We say that 1 is the identity element for multiplication. In addition, 0 is the identity element, because x + 0 = x. In conjunctions, true is the identity element, because x and true = x. In disjunctions, false is the identity element, because x or false = x.

Similarly, in multiplication, x • 0 = 0. We call this the zero property of multiplication, but what is the general term for it? I.e., what do I call it for conjunctions, where x and false = false, and disjunctions, where x or true = true?

Ben Hocking
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    Not sure if this is widespread - but I might suggest something along the lines of "absorbing element". So, when $x \cdot 0 = 0$, the picture would be that 0 "absorbs" the $x$ term. – Daniel Schepler Oct 11 '18 at 00:17
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    @DanielSchepler — Wikipedia thinks it's the correct term: https://en.wikipedia.org/wiki/Absorbing_element Turn that into an answer, and I'll accept it. – Ben Hocking Oct 11 '18 at 01:52

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The generic term is absorbing element, but in semigroup theory, this term is never used and such an element is called a zero.

More precisely, let $S$ be a semigroup. An element $z$ of $S$ such that $xz = z= zx$ for all $x \in S$ is called a zero. It is called a left zero if $z = zx$ for all $x \in S$ and a right zero if $z = xz$ for all $x \in S$.

If a semigroup has a zero, then this zero is unique (this is actually true for magmas). However, a semigroup may have several right zeros or several left zeros, but if a semigroup (or simply a magma) has a right zero and a left zero, then it has a zero.

J.-E. Pin
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