結合法則
最大公約数をとる演算および最小公倍数をとる演算は結合的である: gcd  ( gcd  ( x , y ) , z ) = gcd  ( x , gcd  ( y , z ) ) = gcd  ( x , y , z ) lcm  ( lcm  ( x , y ) , z ) = lcm  ( x , lcm  ( y , z ) ) = lcm  ( x , y , z ) ( ∀ x , y , z ∈ Z ) . {\displaystyle {\begin{array}{l}\operatorname {gcd} (\operatorname {gcd} (x,y),z)=\operatorname {gcd} (x,\operatorname {gcd} (y,z))=\operatorname {gcd} (x,y,z)\\\operatorname {lcm} (\operatorname {lcm} (x,y),z)=\operatorname {lcm} (x,\operatorname {lcm} (y,z))=\operatorname {lcm} (x,y,z)\end{array}}\qquad (\forall x,y,z\in \mathbb {Z} ).}
集合の交叉および合併: ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) = A ∩ B ∩ C ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) = A ∪ B ∪ C ( ∀ A , B , C ) .
( ( P ∨ Q ) ∨ R ) ↔ ( P ∨ ( Q ∨ R ) ) {\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}
( P ∨ ( Q ∨ R ) ) ↔ ( ( P ∨ Q ) ∨ R ) {\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}