Title: Most Complex Non-Returning Regular Languages,
Abstract: A regular language $L$ is non-returning if in the minimal deterministic
finite automaton accepting it there are no transitions into the initial state.
Eom, Han and Jirásková derived upper bounds on the state complexity of
boolean operations and Kleene star, and proved that these bounds are tight
using two different binary witnesses. They derived upper bounds for
concatenation and reversal using three different ternary witnesses. These five
witnesses use a total of six different transformations. We show that for each
$n\ge 4$ there exists a ternary witness of state complexity $n$ that meets the
bound for reversal and that at least three letters are needed to meet this
bound. Moreover, the restrictions of this witness to binary alphabets meet the
bounds for product, star, and boolean operations. We also derive tight upper
bounds on the state complexity of binary operations that take arguments with
different alphabets. We prove that the maximal syntactic semigroup of a
non-returning language has $(n-1)^n$ elements and requires at least
$\binom{n}{2}$ generators. We find the maximal state complexities of atoms of
non-returning languages. Finally, we show that there exists a most complex
non-returning language that meets the bounds for all these complexity measures.