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An M-Set is a unary algebra $\langle X,M\rangle$ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M\rangle$ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M\rangle$ if the congruence lattice of $\langle X,M\rangle$ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\Pi \left(L\right)$ is introduced here. $\Pi \left(L\right)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\Pi \left(L\right)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi$-product lattice. A $\Pi$-product...