Induction–recursion and initial algebras

Induction-recursion and initial algebras

Induction-recursion is a powerful definition method in intuitionistic type theory. It extends (generalized) inductive definitions and allows us to define all standard sets of Martin-Löf type theory as well as a large collection of commonly occurring inductive data structures. It also includes a variety of universes which are constructive analogues of inaccessibles and other large cardinals belo...

Dualising Initial Algebras

Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we s...

Initial Algebras of Determinantal Rings

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.

Subdirectly Irreducible Modal Algebras and Initial Frames

This paper is centred around a nice conjecture, known to Wolfgang Rautenberg and myself since 1978: a modal algebra A is subdirectly irreducible if and only if its dual frame A∗ is initial, or generated (see definitions in the text). I here show that in full generality the conjecture is false, but that it becomes true under some mild additional assumptions. Unfortunately, some counterexamples s...

Infinitary Initial Algebra Specifications for Stream Algebras