Definition. Commutativity [dm-000E]

Liam O'Connor

A binary operator $\bullet$ is commutative iff, for all $x$ and $y$ , $x \bullet y = y \bullet x$ .

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Definition. Commutative monoid [dm-000P]

Liam O'Connor

A commutative monoidis a monoid $(S, \iota , \bullet )$ where the binary operator $(\bullet ) : S \times S \rightarrow S$ additionally satisfie commutativity.

Corollary. Properties of $\sqcup$ [dt-0018]

Liam O'Connor

When lubs exist, the binary $\sqcup$ operator is idempotent, commutative, and associative. Also, we have that $x \sqsubseteq y$ iff $x \sqcup y = y$ .