References

Rieger and Nishimura introduced this construction to study free Heyting algebras on one generator. In particular, the ladder displayed here is the Esakia dual to this algebra.

K. Fine introduced this frame to study incompleteness of S4 logics. V. Shehtman later adapted it to obtain an incomplete superintuitionistic logic. It has served as a useful source of counterexamples: for instance, G. Bezhanishvili, D. Gabelaia, and M. Jibladze employed it in their negative solution to Kuznetsov’s problem for bi-Heyting algebras.

The dual algebras were introduced to study pre-locally tabular superintuitionistic logics and to prove that there are continuum many of them.

This frame was introduced in our studies of pre-local tabularity and local tabularity, seeking a width-bounded locally tabular logic: it is not locally tabular, but all finitely generated algebras of bounded width in its variety are finite. Every finitely generated algebra in this variety is co-Hopfian.

Young's lattice of partitions was introduced by A. Young in the study of integer partitions in combinatorics and number theory. This particular construction models the Fine general frame, and replaces its "kernel" (the bottom ascending part) by Young's lattice of partitions, obtaining interesting properties.

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Bellissima’s universal model gives a representation of the free Heyting algebra on two generators by building the "image-finite part" of the dual space. The construction is recursive, and proceeds by layers, taking into account the structure of successors and the "colors" -- the propositional variables satisfied by each point.

Esakia duality presents a spatial and combinatorial way of representing Heyting algebras, using special compact ordered spaces. Other dualities exist, which employ other representations, such as Bellissima’s, and Ghilardi’s finitely presented duality.