zero elements

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.∎

单词	ZeroElements
释义	zero elements 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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