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Let $S$ be a semigroup. An element $z$ is called a right zero [resp. left zero] if $xz = z$ [resp. $zx = z$ ] for all $x \in S$ .
An element which is both a left and a right zero is called a zero element.
A semigroup may have many left zeros or right zeros, but if it has at least one of each, then they are necessarily equal, giving a unique (two-sided) zero element.
More generally, these definitions and statements are valid for a groupoid.
It is customary to use the symbol $\theta$ for the zero element of a semigroup.
Proposition 1 If a groupoid has a left zero $0_L$ and a right zero $0_R$ , then $0_L = 0_R$ .
Proof. $0_L=0_L 0_R = 0_R$ . 
Proposition 2 If $0$ is a left zero in a semigroup $S$ , then so is $x0$ for every $x\in S$ .
Proof. For any $y\in S$ , $(x0)y=x(0y)=x0$ . As a result, $x0$ is a left zero of $S$ . 
Proposition 3 If $0$ is the unique left zero in a semigroup $S$ , then it is also the zero element.
Proof. By assumption and the previous proposition, $x0$ is a left zero for every $x\in S$ . But $0$ is the unique left zero in $S$ , we must have $x0=0$ , which means that $0$ is a right zero element, and hence a zero element by the first proposition. 
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