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Quotient Complexities of Atoms in Regular Ideal Languages

A (left) quotient of a language $L$ by a word $w$ is the language $w^{-1}L=\{x\mid wx\in L\}$. The quotient complexity of a regular language $L$ is the number of quotients of $L$; it is equal to the state complexity of $L$, which is the number of states in a minimal deterministic finite automaton accepting $L$. An atom of $L$ is an equivalence class of the relation in which two words are equivalent if for each quotient, they either are both in the quotient or both not in it; hence it is a non-empty intersection of complemented and uncomplemented quotients of $L$. A right (respectively, left and two-sided) ideal is a language $L$ over an alphabet $Σ$ that satisfies $L=LΣ^*$ (respectively, $L=Σ^*L$ and $L=Σ^*LΣ^*$). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.