a = a ⊥⊥ {\displaystyle a=a^{\perp \perp }} ∪ {\displaystyle \cup } is commutative and associative. There is a maximal element 1 {\displaystyle 1} , and 1 = b ∪ b ⊥ {\displaystyle 1=b\cup b^{\perp }} for any b {\displaystyle b} . a ∪ ( a ⊥ ∪ b ) ⊥ = a {\displaystyle a\cup (a^{\perp }\cup b)^{\perp }=a} .