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We relate two measures of complexity of regular languages. The first issyntactic complexity, that is, the cardinality of the syntactic semigroup ofthe language. That semigroup is isomorphic to the semigroup of transformationsof states induced by non-empty words in the minimal deterministic finiteautomaton accepting the language. If the language has n left quotients (itsminimal automaton has n states), then its syntactic complexity is at most n^nand this bound is tight. The second measure consists of the quotient (state)complexities of the atoms of the language, where atoms are non-emptyintersections of complemented and uncomplemented quotients. A regular languagehas at most 2^n atoms and this bound is tight. The maximal quotient complexityof any atom with r complemented quotients is 2^n-1, if r=0 or r=n, and1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} \binom{h}{n} \binom{k}{h}, otherwise. Weprove that if a language has maximal syntactic complexity, then it has 2^natoms and each atom has maximal quotient complexity, but the converse is false.

Maximal Syntactic Complexity of Regular Languages Implies Maximal Quotient Complexities of Atoms