Reference request: An algebraic characterisation of LTL[XF]-definable word languages I'm looking for a reference to the fact that LTL[XF]-definable languages (LTL where only the (strict) finally/future modality is allowed) correspond to the variety $\mathbf{R}$ (see: 1).
A similar characterisation is available for LTL[XF,XP], namely the variety $\mathbf{DA}$, see: Theorem 11 from 2.
1 Brzowoski, Fich: Languages of R-Trivial Monoids LINK
2 Tesson, P., Thérien, D.: Diamonds are forever: the variety DA LINK
PS: I have an idea how to prove it (by employing the correspondence that R = partially-ordered DFA), but before writing the result it would make sense to check whether this is already known in the literature (although I do not claim any breakthrough).
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.