a semilattice is a commutative band

This note explains how a semilattice is the same as a commutative band.

Let $S$ be a semilattice, with partial order $<$ and each pair of elements $x$ and $y$ having a greatest lower bound $x\\wedge y$ .Then it is easy to see that the operation $\\wedge$ defines a binary operation on $S$ which makes it a commutative semigroup, and that every element is idempotent since $x\\wedge x=x$ .

Conversely, if $S$ is such a semigroup, define $x\\leq y$ iff $x=xy$ . Again, it is easy to see that this defines a partial order on $S$ , and that greatest lower bounds exist with respect to this partial order, and that in fact $x\\wedge y=xy$ .

单词	ASemilatticeIsACommutativeBand
释义	a semilattice is a commutative band This note explains how a semilattice is the same as a commutative band. Let $S$ be a semilattice, with partial order $<$ and each pair of elements $x$ and $y$ having a greatest lower bound $x\\wedge y$ .Then it is easy to see that the operation $\\wedge$ defines a binary operation on $S$ which makes it a commutative semigroup, and that every element is idempotent since $x\\wedge x=x$ . Conversely, if $S$ is such a semigroup, define $x\\leq y$ iff $x=xy$ . Again, it is easy to see that this defines a partial order on $S$ , and that greatest lower bounds exist with respect to this partial order, and that in fact $x\\wedge y=xy$ .
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