We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially (lattice-ordered) set. Bimonoids form an appropriate framework for the study general notion complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses monoids. The central question paper is whether how bimonoids can be embedded...

We present a game semantics forMartin-Löf type theory (MLTT) that interprets propositional equalities in a non-trivial manner in the sense that it refutes the principle of uniqueness of identity proofs (UIP) for the first time as a game semantics in the literature. Specifically, each of our games is equipped with (selected) invertible strategies representing (computational) proofs of (intension...

We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-albegras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong am...

Usual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that ts intuitionistic logic: Heyting algebras are a more algebraic alternative. In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm fo...

The Brouwer-Heyting-Kolmogorov interpretation of intuition-istic logic suggests that p q can be interpreted as a computation that given a proof of p constructs a proof of q. Dually, we show that every nite canonical model of q contains a nite canonical model of p. If q and p are interderivable, their canonical models contain each other. Using this insight, we are able to characterize validity i...

The variety of Heyting algebras has a nice property that HS = SH. Heyting algebras are the algebraic dual of intuitionistic descriptive frames. The goal of this paper is to define proper dual notions so as to formulate this algebraic properties in the frame language, and to give a frame-based proof of this property and some other duality theorems.

This paper aims to introduce fuzzy congruence relations over Heyting algebras (HA) and give constructions of quotient Heyting algebras induced by fuzzy congruence relations on HA. The Fuzzy First, Second and Third Isomorphism Theorems of HA are established. MSC: 06D20, 06D72, 06D75.