Brzozowski, JanuszDavies, Sylvie2017-10-052017-10-052015http://dx.doi.org/10.14232/actacyb.22.2.2015.4http://hdl.handle.net/10012/12531A (left) quotient of a language L by a word w is the language w(-1) L = {x vertical bar wx is an element of 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 Sigma that satisfies L = L Sigma* (respectively, L = Sigma*L and L = Sigma*L Sigma*). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.enatomleft idealquotientquotient complexityregular languageright idealstate complexitysyntactic semigrouptwo-sided idealArticle