We study short positive (\btie, containing no inverses) identities of finite symmetric groups. The interest in such identities is inspired by the problem of separating words with finite automata, in particular, with the automata in which each letter acts on the set of states as a permutation. We list all positive identities in symmetric groups $S_5$ and $S_6$ up to length 40 and prove that the shortest identity in the full transformation semigroup $T_5$ has length 48. Then we propose the classification of positive identities and describe the identities from some particular classes. We state a conjecture that the shortest positive identity in a sufficiently big symmetric group is ``numerical'', which implies an $O\big(\frac{\log^2 n}{\log\log n}\big)$ upper bound on the words separation function.

Journal of Automata, Languages and Combinatorics, Volume 26, Numbers 1-2, 2021, 67-89