Subloop whose left cosets are pairwise disjoint

Definition
A subloop whose left cosets are pairwise disjoint, or equivalently, a subloop with well-defined left cosets, is a subloop $$S$$ of an algebra loop $$(L,*)$$ satisfying the following equivalent conditions:


 * 1) $$a * (b * S) = (a * b) * S$$ for all $$a,b \in L$$.
 * 2) The sets $$a * S$$, for $$a \in L$$, form a partition of $$L$$.
 * 3) Given $$a,b \in L$$, either $$a * S = b * S$$ or they are pairwise disjoint.
 * 4) The relation $$a \sim b \iff a \in b * S$$ is an equivalence relation on $$L$$.

Note that because left cosets partition a group, any subgroup of a group is a subloop whose left cosets are pairwise disjoint.

Stronger properties

 * Weaker than::Normal subloop

Weaker properties

 * Stronger than::Subloop

Related properties

 * Subloop whose right cosets are pairwise disjoint