Reference request: An algebraic characterisation of LTL[XF]-definable word languages
An algebraic characterisation of the restricted temporal logic (fragment using only next and eventually, but not until) is given in [1].
The expressive power of this fragment is the set of regular languages whose syntactic monoid are locally $\cal L$-trivial.
I am not sure of the result you are looking for, but this article probably contains enough material to help you.
[1] J. Cohen, D. Perrin and J.-É. Pin, On the expressive power of temporal logic for finite words, J. Comput. System Sci. 46 (1993), 271-294.