Equivalence of Definitions of Meet Semilattice

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Theoremm

The following definitions of the concept of Meet Semilattice are equivalent:

Definition 1

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

$a \wedge b \in S$,

where $a \wedge b$ is the meet of $a$ and $b$.

Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a meet semilattice.

Definition 2

Let $\struct {S, \wedge}$ be a semilattice.

Let $\preceq$ be the ordering on $S$ defined by:

$a \preceq b \iff \paren {a \wedge b} = a$

Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a meet semilattice.

Proof

Definition 1 of meet semilattice is the dual statement of definition 1 of join semilattice by Dual Pairs (Order Theory).

Definition 2 of meet semilattice is the dual statement of definition 2 of join semilattice by Dual Pairs (Order Theory).

Hence this theorem is the dual statement of Equivalence of Definitions of Join Semilattice by Dual Pairs (Order Theory).

The result follows from the Duality Principle.

$\blacksquare$