Join Operation on Ordered Set such that Every Doubleton admits Supremum is Entropic

Theorem

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

Let $\struct {S, \preccurlyeq}$ be such that every doubleton subset of $S$ admits a supremum.

Let $\vee$ be the join operation on $S$, defined as:

$\forall a, b \in S: a \vee b = \sup_\preccurlyeq \set {a, b}$

Then $\vee$ is an entropic operation.

Proof

By definition, $\struct {S, \vee, \preccurlyeq}$ is a join semilattice.

From Join Semilattice is Semilattice, $\struct {S, \vee, \preccurlyeq}$ is indeed a semilattice.

Then:

\(\ds \forall a, b, c, d, \in S: \, \)

\(\ds \paren {a \vee b} \vee \paren {c \vee d}\)

\(=\)

\(\ds a \vee \paren {\paren {b \vee c} \vee d}\)

Semilattice Axiom $\text {SL} 1$: Associativity

\(\ds \)

\(=\)

\(\ds a \vee \paren {\paren {c \vee b} \vee d}\)

Semilattice Axiom $\text {SL} 2$: Commutativity

\(\ds \)

\(=\)

\(\ds \paren {a \vee c} \vee \paren {b \vee d}\)

Semilattice Axiom $\text {SL} 1$: Associativity

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.24$