Preceding iff Meet equals Less Operand

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Theorem

Let $\left({S, \preceq}\right)$ be a meet semilattice.

Let $x, y \in S$.

Then

$x \preceq y$ if and only if $x \wedge y = x$


Proof

Sufficient Condition

Let

$x \preceq y$

By definition of meet:

$x \wedge y = \inf \left\{ {x, y}\right\}$

By definitions of lower bound and reflexivity:

$x$ is lower bound for $\left\{ {x, y}\right\}$

and

$\forall z \in S: z$ is lower bound for $\left\{ {x, y}\right\} \implies z \preceq x$

Thus by definition of infimum:

$x = \inf \left\{ {x, y}\right\} = x \wedge y$

$\Box$

Necessary Condition

Let

$x \wedge y = x$

Thus by Meet Precedes Operands:

$x \preceq y$

$\blacksquare$


Sources