A local proof for a tolerance intersection property (Q2577766)

The identity \(T^*\cap S^*=(T\cap S)^*\), where \(T,S\) are tolerances and \((\;)^*\) denotes transitive closure, is called Tolerance Intersection Property (TIP). The identity \(T^*\cap S^*=(T\cap(S\circ S))^*\) is named weak Tolerance Intersection Property (wTPI\(_2)\). TIP has a number of important consequences, namely it known that any congruence modular variety satisfies TIP. Now the author proves that an algebra \(A\) has wTIP\(_2\) whenever all subalgebras of the power \(A^4\) satisfy the Shifting Lemma.