# 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.