Introductory Text On Heyting Algebras

Question

I am looking for an introductory text on Heyting algebras, and specifically their relation logic. Searching the internet I found Heyting Algebras: Duality Theory by Leo Esakia, but do not know whether this is the best text with which to begin. I scanned through all other questions on the mathematics stack exchange with the heyting-algebra tag looking for references to books, but found only books in which Heyting algebras are mentioned tangentially or in reference to some other topic. I also found a way to access what I believe are Heyting's original papers (Die formalen Regeln der intuitionistischen Logik. I, II, III), but in German, and can't find a translation. English translations of these would also be interesting. As a note, my experience with algebra is limited to an undergraduate course on abstract algebra covering groups and rings, as well as an upper-division undergraduate course on group representation theory (in which algebras were introduced). A book accessible at this level is preferred, though I am willing to work/self-teach in order to understand a higher level reference.

Answer 1

The only book fully about Heyting algebras I do know is the one by Leo Esakia that you already mentioned. It might be a bit advanced if all you are looking for is an introduction to the subject, but please keep in mind that it is the standard reference if you are interested in the problem of representation of Heyting algebras and in so-called Esakia duality.

In general, I recommend looking at the following books/notes. They cover way more than Heyting algebras only, but I am sure you will find some sections useful.

• H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics: this is a classic old book that covers propositional logics and lattice-based algebras with great depth. Just keep in mind that here Heyting algebras are called Pseudo-Boolean algebras.

• A. Chagrov and M. Zakharyaschev, Modal Logic: this book covers has a lot of material about propositional modal and intermediate logics, and it also covers their algebraic semantics. In particular, look at the part on Pseudo-Boolean algebras in the chapter on the algebraic semantics of intuitionistic and intermediate logics.

• D. De Jongh and Nick Bezhanishvili, Intuitionistic Logic: these notes have been circulating for a while and they could be very useful. Again, the starting point is intuitionistic logic, but they treat Heyting algebras extensively.

Also General lattice theory by ‎Grätzer and A course in Universal Algebra by Burris and Sankappanavar can be useful reference, but they do not focus on Heyting algebras specifically. Finally, if your interest is somehow set-theoretical, Set Theory. Boolean Valued Models and Independence Proofs by John Bell has some chapters dedicated to Intuitionistic Set Theory and Heyting-Valued Models.