Quotient Complexities of Atoms in Regular Ideal Languages

A (left) quotient of a language L by a word w is the language wL = {x | wx ∈ 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 = ΣL and L = ΣLΣ). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.