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?
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.