Let \(\mathbf M\) be a monoid of transformations on a set \(A\). An imprimitivity relation of \(\mathbf M\) is an equivalence relation \(\varepsilon\) on \(A\) such that \((\phi(x), \phi(y)) \in \varepsilon\) for all \(x, y \in X\) and \(\phi \in {\mathbf M}\). Let \(\varepsilon\) be an imprimitivity relation of \(\mathbf M\). Then \(\varepsilon = \bigcup_{\alpha \in \lambda} (A_\alpha \times A_\alpha)\) where \(\{A_\alpha\}_{\alpha \in \lambda}\) is the decomposition of \(A\) which is induced by \(\varepsilon\). A part of \(\varepsilon\) is any relation of the form \(\vartheta = \bigcup_{\alpha \in \mu} (A_\alpha \times A_\alpha)\) where \(\mu \subseteq \lambda\) and the set \(\bigcup_{\alpha \in \mu} A_\alpha\) is denoted by \(pr\vartheta\). The relation \(\vartheta\) is said to be conditional if it uniquely determines \(\varepsilon\). A relation \(\tau_M\) (which is a bit too complicated to give here) is then defined on \(A \times A\) and in the main theorem, the author verifies that if \((A \setminus X) \times (A \setminus X) \subseteq \tau_M (X \times X)\) where \(X = pr \vartheta\), then \(\vartheta\) is conditional. The author then applies this result and obtains various other results concerning semigroups and lattices.