# Questions tagged [heyting-algebra]

This tag is for questions about Heyting algebras, which are lattices with certain properties, generalizing the concept of boolean algebras. This tag may be used for questions about algebraic semantics for [tag:intuitionistic-logic]. For more general questions about lattices use [tag:lattice-orders]. For more specific questions about boolean algebras, use [tag:boolean-algebra].

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### Do the monotone maps from a poset into a Heyting algebra form a Heyting algebra?

I am interested in generalizing the fact that the up-sets of a poset always form a Heyting algebra. Let $P$ be a poset and $H$ a Heyting algebra. $\operatorname{Hom}(P,H)$ can be made a bound lattice ...
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### A weaker version of the De Morgan algebras

A De Morgan algebra is a structure $\langle A, \lor, \land,0,1,\neg \rangle$ such that $\langle A, \lor, \land,0,1 \rangle$ is a bounded distributive lattice and $\neg$ is a involution that ...
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### What can be said about a Bi-Heyting algebra when the complement operations are useless?

Imagine, for whatever reason, a bounded lattice[1] $(L, 0, 1, ∧, ∨)$ which is a bi-heyting algebra, i.e. there is an operation $→$ such that $x∧y ≤ z$ iff $x ≤ y→z$ (the heyting algebra structure) and ...
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### Identity in Heyting algebras or not

In some computation over a Heyting algebra, I ended up with the following formula: $$\Big[(x\to y)\to z\Big]\to \Big[\big(x\to(y\vee z)\big)\vee \big((x\to(y\vee z))\to z\big)\Big]$$ I wonder if it is ...
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### Books on co-Heyting algebras (with a view to their logics).

I would like to know more about co-Heyting algebras, particularly from the perspective of their logics (like paraconsistent logics). What books are available out there on the topic? It might be that ...
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### Techniques for showing that an equational theory like Heyting algebras is "group-free"

One thing that's interesting about classical propositional logic is the presence of a group inside it. In any Boolean ring, the XOR $\oplus$-reduct is a Boolean group. If we take a Heyting algebra, ...
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### Definition of valuation of propositions in an Heyting algebra

What is the standard definition of valuation of propositions in an Heyting algebra? A valuation of propositions of a propositional language in an Heyting algebra $(H, \wedge, \vee, \rightarrow, 1, 0)$ ...
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### Help calculating relative pseudo-complements in a Heyting algebra/lattice

I'm trying to work some examples of relative pseudo-complements in lattices, to make sure I understand them. I wonder if anybody could check my examples, and tell me if I'm correct or if I've ...
1 vote
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### Proof that regular elements of Heyting algebra form a Boolean algebra

Let H be a Heyting algebra with join operation v. It's a well-known fact that the regular elements of H form a Boolean algebra if v is redefined as (a, b) -> (a V b)** However, I can't find this ...
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### Is every Heyting algebra a sublattice of a Boolean algebra?

From what I can tell, every lattice is a sublattice of a lattice with unique complements (Dilworth). A Heyting algebra is a distributive lattice. The only remaining step, then, would be to know ...
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### Question on Heyting algebras

Does $a \Rightarrow b = 1$ iff $a≤b$ hold for any complete Heyting algebra? If not, please provide a counterexample.
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### Intermediate logics and strong algebraic completeness

As a setup, suppose that you have a usual propositional language $\mathcal L$ over a set of propositional variables $Var$ and with symbols $\land,\lor,\rightarrow,\bot$ in the usual way. Let $L$ be an ...
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### Interpretations of Topological Space as a Heyting Algebra

I have recently learned about Heyting algebras which I find quite fascinating, as I am more intuitionistically inclined. One of the main examples of Heyting algebras are given by topological spaces as ...
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### Proving $\Gamma \vdash \phi$ Implies $\Gamma \vDash \phi$ (for Institutionistic propositional logic and Heying algebras)

I'm trying to prove that $\Gamma \vdash \phi$ implies $\Gamma \vDash \phi$ (for Institutionistic propositional logic and Heying algebras), by induction with respect to natural deduction proofs of ...
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### Does every intuitionistic formula have disjunctive or conjuctive normal form?

As in title - does every intuitionistic formula have disjunctive and conjuctive normal form? I guess that this is correct but I couldn't find any information on that.
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### Cocompleteness of the category of $H$-sets.

Let $H$ be a complete Heyting algebra, i.e, a complete poset which is also cartesian closed as a category. In particular, this algebra has a least and a greatest element. We can define the category of ...
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### Does $\mathbf{N}$ with the reverse divisibility order form a Heyting algebra?

Consider the nonnegative integers $\mathbf{N}$ with the reverse divisibility order (i.e. $\mathrm{a} \leq \mathrm{b}$ $\iff$ $\mathrm{b} \mid \mathrm{a}$). Is this a Heyting algebra? One advantage ...
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### Example of a finite Heyting algebra that is not Boolean

Simple question: what are some simple examples of a finite Heyting Algebras, that is not also a Boolean Algebra?
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### About the complement of a subobject in a topos

Let $\mathcal{E}$ be a topos and let $X$ be an object of $\mathcal{E}$. Let $S \to X$ be a subobject of $X$. We only know that the category of the subobject of $X$ is a Heyting Algebra, so we do not ...
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### Does the frame of open sets in a topological space or locale really have all meets?

According to the nLab article on locales, a frame has all meets by the adjoint functor theorem: This seems a bit strange to me, since it's well-known that an infinite intersection of open subsets is ...
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### Inverse of Heyting algebra morphism is p-morphism

It is known that the category of Heyting algebras is dually equivalent to the category of Esakia spaces (which is equivalent to the category of descriptive intuitionistic Kripke frames). Under the ...
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### Relative consistency of ZF with respect to IZF

Is there a forcing argument of this fact? Can anybody point me to the place? The reason I'm asking is because I was reading Heyting-Valued Models for Intuitionistic Set Theory by R.J. Grayson, yet the ...
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### Is $\mathsf{HA}$ the ind-completion of $\mathsf{FinHA}$

We know that the category of Boolean algebras and homomorphisms is the ind-completion of $\mathsf{FinBA}$, the full subcategory of $\mathsf{BA}$ of finite Boolean algebras. I am wondering if the same ...
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### Characterization of zero-dimensional frames via lattices of ideals

My question concerns the left-to-right implication of the following: Theorem A frame $L$ is compact and zero-dimensional iff it is isomorphic to the lattice of ideals $\mathcal{I}(B)$ of some Boolean ...
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### What is the intuitive meaning of the Heyting algebra?

I'm curious about this. As we all know, classical logic can be described as what is called the Boolean Algebra, in particular, you can think of figuring out the truth of a classical composite ...
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### What does "weakest proposition" mean in Heyting algebra?

So I was reading the Wikipedia page about Heyting algebra. What does "weakest proposition" (formally) mean in Heyting algebra? In mathematics, a Heyting algebra is a bounded lattice (with join and ...
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Let $f\colon X \longrightarrow Y$ a continuous function between topological spaces. This induces a frame morphism $f^{-1}:\tau_Y\longrightarrow \tau_X$. I'm looking for an example when $f^{-1}$ is not ...