Subhomomorphism relation between transfer condition operator and intersection operator

Statement

Suppose $\cap$ denotes the intersection operator on subgroup metaproperties and $T$ denotes the Transfer condition operator (?). Then, we have:

$\!T(p)\cap T(q)\leq T(p\cap q)$.

Related facts

• Transfer condition operator preserves intersection-closedness
• Intersection-closure operator preserves transfer condition