$T$-partitions of quasigroups and groups

We introduce the concept of a $T$-partition of a quasigroup that generalizes various situations when the Cayley table of a quasigroup can be partitioned into smaller Latin squares. From all the $T$-partitions we identify left-regular [resp. right-regular], regular and homogeneous $T$-partitions. The $T$-partitions of each type of quasigroup form a lattice. We study the lattices of $T$-partitions of groups. In particular, we prove that any finite abelian group can be uniquely reconstructed up to isomorphism from the lattice of its left-regular [resp. right-regular] $T$-partitions.