Structure Induced by Semilattice Operation is Semilattice

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Theorem

Let $\struct {T, \circ}$ be a semilattice.

Let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Then $\struct {T^S, \oplus}$ is a semilattice.

Proof

Taking the semilattice axioms in turn:

Semilattice Axiom $\text {SL} 0$: Closure

As $\struct {T, \circ}$ is a semilattice, it is closed by Semilattice Axiom $\text {SL} 0$: Closure.

From Closure of Pointwise Operation on Algebraic Structure it follows that $\struct {T^S, \oplus}$ is likewise closed.

$\Box$

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

As $\struct {T, \circ}$ is a semilattice, $\circ$ is a fortiori associative.

So from Structure Induced by Associative Operation is Associative, $\struct {T^S, \oplus}$ is also associative.

$\Box$

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

As $\struct {T, \circ}$ is a semilattice, $\circ$ is a fortiori commutative.

So from Structure Induced by Commutative Operation is Commutative, $\struct {T^S, \oplus}$ is also commutative.

$\Box$

Semilattice Axiom $\text {SL} 3$: Idempotence

As $\struct {T, \circ}$ is a semilattice, $\circ$ is a fortiori idempotent.

So from Structure Induced by Idempotent Operation is Idempotent, $\struct {T^S, \oplus}$ is also idempotent.

$\Box$

All the semilattice axioms are thus seen to be fulfilled, and so $\struct {T^S, \oplus}$ is a semilattice.

$\blacksquare$