Equivalence of Definitions of Join Semilattice

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Theoremm

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

Definition 1

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

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

$a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.

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

Definition 2

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

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

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

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

Proof

Definition 1 implies Definition 2

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

For all $a, b \in S$ let:

$a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.

We have by hypothesis, $\vee$ is closed.

The other defining properties for a semilattice follow respectively from:

Join is Commutative

Join is Associative

Join is Idempotent

Hence $\struct {S, \vee}$ is a semilattice.

From Successor is Supremum:

$\forall a, b \in S : a \preceq b \iff b = a \vee b$

$\Box$

Definition 2 implies Definition 1

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

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

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

From Semilattice has Unique Ordering such that Operation is Supremum:

$\preceq$ is the unique ordering on $S$ such that:

$\forall a, b \in S : a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$

$\blacksquare$