I'm reading up on infinite generalizations of the fundamental theorem of distributive lattices. [Wikipedia](https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem#Generalizations) (June 15, 2017) says that there is a duality 



> 
> between distributive lattices and [coherent spaces](https://en.wikipedia.org/wiki/Coherent_space) (sometimes called [spectral spaces](https://en.wikipedia.org/wiki/Spectral_space))
> 
> 
> 


where I have reproduced the links as they appear. I think that the article on coherent spaces they have linked to is referring to a different, unrelated notion. Could someone in logic check me on this before I correct it?


Thanks, and apologies if this is too frivolous a use of MO.



Is this Wikipedia article linking to the wrong notion of coherent space
Yes, the notion of "coherent space" at the page you linked is completely unrelated to spectral spaces. The coherent spaces of that article were introduced by Jean-Yves Girard as a denotational semantics of second-order intuitionistic logic and were instrumental in his discovery of linear logic.


People have been aware of this potential confusion since the beginning of linear logic (see footnote 3, p.55 of Girard, Lafont and Taylor's book *Proofs and Types*), that is why some prefer to call them *coherence spaces*. I guess that the authors of the Wikipedia page didn't follow this practice.


**Edit:** Just after posting my answer, I realized that this potential confusion is explicitly pointed out in the [Wikipedia page](https://en.wikipedia.org/wiki/Coherent_space), as well as the alternative terminology in *Proofs and Types*.


