It seems well known in modal logic society that $\Box(\Box p\to p) \to \Box p$ in Kripke semantics of $GL$ implies well-foundedness of the relation i.e. no infinite ascending chains are allowed.


And validity of $GL$ additionally implies irreflexivity and transitivity of the relation.


But I am wondering what is the initial-like reference for the facts mentioned.


**P.S.** This question may not make clear anything interesting, this is just reference request to experts.


**UPDATE** It seems that Segerberg's thesis is tha earliest reference for the fist fact.


Concerning the second fact I reckon Dick de Jongh has proved this one, that said, unfortunately I could not find any reference.



Initial reference on Gödel-Löb axiom in Kripke semantic of $GL$
The paper [*Provability: the emergence of a mathematical modality*](https://www.jstor.org/stable/20015550) by Boolos and Sambin says (bottom of page $9$) that the first fact was independently gotten by Kripke and Segerberg, and gives the latter's $1971$ thesis as a reference. However, I can't find a copy of Segerberg's thesis online; the only promising link I found, [http://lpcs.math.msu.su/~zolin/ml/pdf/Segerberg\_Essay\_1971.pdf](http://lpcs.math.msu.su/%7Ezolin/ml/pdf/Segerberg_Essay_1971.pdf), seems broken.


The second fact you mention has trivial first conjunct and false second conjunct (non-transitive frames may still be well-founded), and so I doubt there is a citation for it.


