Subdistributivity relation between intersection operator and composition operator

Statement

Finite version

Suppose $p_1,p_2,q$ are subgroup properties. Let $\cap$ denote the Intersection operator (?) on subgroup properties, and $*$ denote the Composition operator (?) on subgroup properties. Then:

$(p_1\cap p_2)*q\le (p_1*q)\cap (p_2*q)$.

Related facts

Corollaries

• Finite-intersection-closedness is left residual-closed: This is a corollary of the finite version.
• Intersection-closedness is left residual-closed: This is a corollary of the infinite version.