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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 ...
user4614475's user avatar
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0 answers
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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 ...
MtSet's user avatar
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1 vote
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51 views

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 ...
Lukas Juhrich's user avatar
2 votes
1 answer
74 views

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 ...
Evgeny Kuznetsov's user avatar
1 vote
0 answers
58 views

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 ...
Shaun's user avatar
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1 vote
1 answer
67 views

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, ...
Greg Nisbet's user avatar
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1 vote
0 answers
29 views

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 $(H, \wedge, \vee, \rightarrow, 1, 0)$ ...
effezeta's user avatar
  • 455
7 votes
0 answers
236 views

Complements of subobjects and algebraic geometry

This is an idle question, I don't really have any particular application in mind. In any category $C$, for any object $c \in C$ we can consider its subobjects, namely monomorphisms $d \hookrightarrow ...
Qiaochu Yuan's user avatar
2 votes
1 answer
83 views

Intuitionistic proof of $((p\rightarrow q)\rightarrow p)\rightarrow\neg \neg p$

I need to prove that the $\psi=((p\rightarrow q)\rightarrow p)\rightarrow\neg \neg p$ is intuitionistically valid. I tried using the topology of open sets of $\mathbb{R}$ and an arbitrary valuation, ...
Νικολέτα Σεβαστού's user avatar
1 vote
0 answers
66 views

What do algebraic semantics look like for intuitionistic modal logic?

I know that a topological pseudo-Boolean algebra is an algebra ⟨L, I, ¬, ∧, ∨, →⟩ such that ⟨L, ¬, ∧, ∨, →⟩ is an algebra and I an interior unary monotone operator on L, where the operator is defined ...
Andres's user avatar
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4 votes
1 answer
109 views

What is the canonical topology on a complete heyting algebra?

I have troubles with catching the dividing line on Grothendieck topologies over posets between subcanonical and not subcanonical. In particular I have the following very specific basic question which ...
curious on mathematics's user avatar
1 vote
1 answer
194 views

Completeness of intuitionistic propositional logic

I want to study the semantics of intuitionistic propositional logic. Fix a topological space $X$, let $\Gamma\models^X A$ denote that $A$ is valid under $\Gamma$ in the algebra of all open sets of $X$....
wxkj99's user avatar
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3 votes
1 answer
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Uniqueness of arrows between Heyting-valued sets

I’m going through Goldblatt’s “Topoi”, in particular through the subchapter 11.9 on Heyting-valued sets, and it feels overly… concise. In short, I’m not sure how to prove uniqueness of arrows. For ...
0xd34df00d's user avatar
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3 votes
1 answer
133 views

left-adjoint to join in a Heyting algebra

Define a Heyting algebra to be a bounded lattice $L$ with an operation $\to : L^{op}\times L \to L$ such that for any $x, a, b \in L$ we have $x\wedge a \leqslant b$ iff $x \leqslant a \to b$. ...
Matthew Towers's user avatar
4 votes
1 answer
153 views

Is there a topos of Heyting algebra valued models?

While reading about forcing theory, I came across the concept of Boolean algebra valued models of ZFC. The main idea seems to be: instead of the sheaf/presheaf approach of focusing on a particular ...
Daniel Schepler's user avatar
3 votes
1 answer
90 views

Is a Hetying algebra with coexponetials Boolean?

Suppose we have a Heyting algebra $\mathcal A$ with coexponentials. Specifically, for every $a, b : \mathcal A$ we have an object $b \backslash a$ with the properties that $b \le a \lor (b \backslash ...
Nate's user avatar
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1 vote
1 answer
143 views

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 ...
user1439929's user avatar
1 vote
1 answer
187 views

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 ...
Paul Epstein's user avatar
4 votes
2 answers
174 views

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 ...
Carcassi's user avatar
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1 answer
105 views

Question on Heyting algebras

Does $a \Rightarrow b = 1$ iff $a≤b$ hold for any complete Heyting algebra? If not, please provide a counterexample.
Eddie Chau's user avatar
6 votes
0 answers
129 views

Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary)

A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\mathcal O$ is a frame, then $\mathrm{Sh}(\mathcal ...
user978360's user avatar
-1 votes
3 answers
104 views

$(p\rightarrow q)\vee (q\rightarrow r)$ imply peirce's law in intuitionistic logic? [closed]

Consider the system of intuitionistic implicational logic together with the axiom schema adding all instances of $$(p\rightarrow q)\vee(q\rightarrow r)$$ Does that make Peirce's law true? I have ...
Paul's user avatar
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1 vote
0 answers
57 views

Modal Heyting Algebras

Is there a standard way to add modal operators over a Heyting algebra -- as it was done e.g. by Johnstone and Tarski for Boolean algebras? Does this provide a semantics to some version of ...
D.Q.'s user avatar
  • 358
1 vote
1 answer
102 views

Details about the relation about interior algebras and Heyting algebras

I was reading the wikipédia article about Interior algebra: https://en.wikipedia.org/wiki/Interior_algebra And I found a passage that made me very curious: "The open elements of an interior ...
Victor's user avatar
  • 289
3 votes
1 answer
312 views

Is every finite distributive lattice a finite Heyting algebra?

It is known that every Heyting algebra is a distributive lattice. I was wondering if the reverse is true in the finite case. That is, every finite distributive lattice is a (finite) Heyting algebra. I ...
PG Segador's user avatar
3 votes
1 answer
406 views

Introductory Text On Heyting Algebras

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 ...
The Ledge's user avatar
  • 215
2 votes
1 answer
58 views

Nice description of the Heyting implication for clopen upsets

By the Priestley duality, we know that a lattice can be represented as the clopen upsets of its prime filters space $X$, namely via the map $$ \eta: a \in L \longmapsto \lbrace x \in X \mid x \ni a \...
Bijco's user avatar
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10 votes
1 answer
1k views

How Do Heyting Algebras Relate To Logic?

My question is, broadly speaking, how are Heyting algebras related to logic ? It would be great if someone could answer this question without being too technical (or point to easy to read literature). ...
Richard Southwell's user avatar
7 votes
1 answer
107 views

Is the quotient of a complete Heyting algebra by a prime filter complete?

Suppose $\mathbf A$ is a complete Heyting algebra and $F$ is a prime filter on $\mathbf A$. Do we then have that the quotient $\mathbf{A}/F$ is complete? I have tried to prove this, but in my proof I ...
blub's user avatar
  • 4,813
2 votes
1 answer
107 views

Equivalence of definitions of Heyting Algebra

I encountered the following problem in Burris' A Course in Universal Algebra: If $\langle H,\vee,\wedge,\rightarrow,0,1\rangle$ is a Heyting algebra and $a,b\in H$ show that $a\rightarrow b$ is the ...
user722227's user avatar
1 vote
1 answer
79 views

Please help me prove that the equational definition of Heyting Semilattice is equivalent to the Order-Theoretic definition

The situation: An algebraic semilattice $(L, \wedge, \top)$ satisfies the following axioms: $\forall x \in L. x \wedge \top = x$ $\forall x, y \in L. x \wedge y = y \wedge x$ $\forall x, y, z \in L. (...
MonadMania's user avatar
5 votes
1 answer
67 views

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 ...
blub's user avatar
  • 4,813
9 votes
1 answer
512 views

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 ...
TheEmptyFunction's user avatar
4 votes
1 answer
113 views

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 ...
Michael Novak's user avatar
0 votes
1 answer
108 views

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.
liew's user avatar
  • 115
1 vote
0 answers
51 views

About a proof that localic maps are open iff their left adjoints are complete Heyting algebra homomorphisms

I'm reading the proof of the statement in the title in Picado and Pultr's book Frames and Locales (Proposition 7.2). The authors first obtain the formula $x\wedge \phi(a) = y\wedge \phi(a)\iff f^*(x)...
greens's user avatar
  • 53
5 votes
1 answer
1k views

How to show the double negation law in Boolean algebra

I want to show the double negation law $\lnot \lnot s = s \tag{0}$ where $s$ is an element of Boolean algebra. And $\lnot$ is defined as $\lnot s := s \rightarrow 0$. Boolean algebra is a Heyting ...
konyonyo's user avatar
  • 189
5 votes
1 answer
241 views

How show the distributivity of Heyting algebra

I want to show the following distributivity of Heyting algebra $x \rightarrow (y \land z) = (x \rightarrow y) \land (x \rightarrow z) \tag{0}$ using only the below four laws $x \rightarrow x = 1 \...
konyonyo's user avatar
  • 189
1 vote
1 answer
57 views

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 ...
EBP's user avatar
  • 1,388
1 vote
1 answer
81 views

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 ...
Geoffrey Trang's user avatar
5 votes
2 answers
613 views

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?
Neuromath's user avatar
  • 648
1 vote
2 answers
280 views

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 ...
Matteo Spadetto's user avatar
6 votes
1 answer
345 views

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 ...
ಠ_ಠ's user avatar
  • 10.9k
1 vote
0 answers
108 views

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 ...
Math Student 020's user avatar
1 vote
0 answers
116 views

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 ...
Christopher Rose's user avatar
3 votes
1 answer
112 views

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 ...
Math Student 020's user avatar
2 votes
1 answer
136 views

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 ...
Rafał Gruszczyński's user avatar
4 votes
1 answer
634 views

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 ...
The_Sympathizer's user avatar
1 vote
1 answer
67 views

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 ...
mavavilj's user avatar
  • 7,296
0 votes
1 answer
47 views

Looking for an example of a continuous function such that the inverse image is not a Heyting Algebras morphism

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 ...
Math.mx's user avatar
  • 1,959