  The words separation problem, originally formulated by Goralcik and Koubek
(1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for
any two words of length $\le n$ there is a deterministic finite automaton with
$Sep(n)$ states, accepting exactly one of them. The problem is to find the
asymptotics of the function $Sep$. This problem is inverse to finding the
asymptotics of the length of the shortest identity in full transformation
semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity
in $T_k$. We find the first series of identities in $T_k$ which are shorter
than the corresponding unary identity for infinitely many values of $k$, and
thus slightly improve the lower bound on $Sep(n)$. Then we present some short
positive identities in symmetric groups, improving the lower bound on
separating words by permutational automata by a multiplicative constant.
Finally, we present the results of computer search for short identities for
small $k$.
