The Order-Theoretic Structure of Free Heyting Algebras
Michael O'Connor
我们发现了n元直觉命题逻辑的Lindenbaum代数的序论刻画。
We find an order-theoretic characterization of the Lindenbaum algebra of intuitionistic propositional logic in n variables.
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A Heyting algebra is supplemented if each element $a$ has a dual pseudo-complement $a^+$, and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally s…
The paper is devoted to an algebraic interpretation of Kuznetsov's theorem which establishes the assertoric equipollence of intuitionistic and proof-intuitionistic propositional calculi. Given a Heyting algebra, we define an enrichable Heyting algebra, in which the former algebra is embedded. Moreover, we show that both algebras generate one and the same variety of Heyting algebras. This algebraic result is equivalent to the Kuznetsov theorem. The proposed construction of the enrichable `counterpart' of a given Heyting algebra allows one to observe that some properties of the original algebra …
The variety DMSH of semi-Heyting algebras with a De Morgan negation was introduced in [12] and an increasing sequence DMSHn of level n, n being a natural number, of its subvarieties was investigated in the series [12], [13], [14], [15], [16], and [17], of which the present paper is a sequel. In this paper, we prove two main results: Firstly, we prove that DMSH1-algebras of level 1 satisfy Stone identity, generalizing an earlier result that regular DMSH1-algebras of level 1 satisfy Stone identity. Secondly, we prove that the variety of DmsStSH of dually ms, Stone semi-Heyting algebras is at lev…
We prove that there exist profinite Heyting algebras that are not isomorphic to the profinite completion of any Heyting algebra. This resolves an open problem from 2009. More generally, we characterize those varieties of Heyting algebras in which profinite algebras are isomorphic to profinite completions. It turns out that there exists largest such. We give different characterizations of this variety and show that it is finitely axiomatizable and locally finite. From this it follows that it is decidable whether in a finitely axiomatizable variety of Heyting algebras all profinite members are p…
Hereditarily Structurally Complete Superintuitionistic Logics and Primitive Varieties of Heyting Algebras
本文通过代数方法证明了中间逻辑的可遗传结构完备性与Heyting代数变种的原始性之间的等价性。
On the preservation of unification type of Heyting algebras and interior algebras
本文研究W. Blok论文中函子对局部有限内部代数簇和Heyting代数簇统一类型保持的影响。
In this paper intuitionistic topological system and its properties have been introduced. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionistic topological systems have also been studied in details.
In this paper, we will study Heyting algebras endowed with tense negative operators, which we call tense H-algebras and we proof that these algebras are the algebraic semantics of the Intuitionistic Propositional Logic with Galois Negations. Finally, we will develop a Priestley-style duality for H-algebras.
We axiomatize the model-completion of the theory of Heyting algebras by means of the "Density" and "Splitting" properties in [DJ18], and of a certain "QE Property" that we introduce here. In addition: we prove that this model-completion has a prime model, which is locally finite and which we explicitly construct; we show how the Open Mapping Theorem of [vGR18] can be derived from the QE Property of existentially closed Heyting algebras; and we construct a certain "discriminant" for equations in Heyting algebras, similar to its ring theoretic counterpart.
Esakia Duals of Regular Heyting Algebras
本文利用 Esakia 对偶刻画正则 Heyting 代数对应的空间，证明其生成连续统多个 Heyting 代数簇，并据此得到探究逻辑、$\mathtt{DNA}$-逻辑和依赖逻辑的新拓扑完备性定理。
On locally finite varieties of Heyting algebras
本文构造了一种Heyting代数簇，证明对任意自然数$n$，其$n$-生成自由代数有限而$(n+1)$-生成自由代数无限。
We provide a recursive construction of all the semi-Heyting algebras that can be defined on a chain with $n$ elements. This construction allows us to count them easily. We also compare the formula for the number of semi-Heyting chains thus obtained to the one previously known.
Discrete Duality for Tense Symmetric Heyting Algebras
本文继续研究了紧张对称Heyting代数（或TSH-代数）。这些代数构成了紧张代数的推广。特别地，我们描述了TSH代数的离散对偶性，同时考虑到E.或者我和洛斯卡。Rewitzky在[E.或者我和洛斯卡。李文，带算子的Heyting代数的离散对偶，基金会。81（2007），no.1-3，275-295.]关于Heyting代数此外，我们还引入了一个命题演算，并证明了这个演算有TSH-代数作为代数对应。最后，上面提到的对偶性使我们能够证明这种微积分的完备性定理。
The variety DQD of semi-Heyting algebras with a weak negation, called dually quasi-De Morgan operation, and several of its subvarieties were investigated in a series of four papers. In this paper we define and investigate a new subvariety JID of DQD, called JI-distributive, dually quasi-De Morgan semi-Heyting algebras, as well as the variety DSt of dually Stone semi-Heyting algebras. We first prove that DSt and JID are discriminator varieties of level 1 and level 2, respectively. Secondly, we give a characterization of subdirectly irreducible algebras of the subvariety JID1 of level 1 of JID. …
Roelcke non-precompactness, non-simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are examined among others.
We show that if a subgroup of the automorphism group of the Fraisse limit of finite Heyting algebras has a countable index, then it lies between the pointwise and setwise stabilizer of some finite set.
On Bellissima's construction of the finitely generated free Heyting algebras, and beyond
本文从拓扑学和模型论的角度研究了非生成的自由Heyting代数。我们审查Bellissima的代表性的生成的自由海廷代数，我们证明，它产生一个嵌入profinite完成，这也是完成相对于一个自然定义的度量。我们给出了一个代数解释的Kripke模型Bellissima作为主理想sprectrum，并表明它是一阶解释的海廷代数，从几个模型的理论和代数性质的衍生。例如，我们证明了一个自由生成的Heyting代数只有一组自由生成元，这组自由生成元是可定义的，从而它的自同构群是它的生成元上的置换群.
The superamalgamation property for reducts of Heyting polyadic algebras with and without equality
我们证明了无限维Heyting多进代数的几个约简，有和没有平等享有各种合并性质。在无等式的情况下，我们得到超合并，但当我们有等式时，我们得到一个较弱的插值性质，相应的无限直觉主义逻辑与等式。
We show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting-Brouwer logic $\mathsf{HB}$ that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from intermediate logics to extensions of $\mathsf{HB}$.
We introduce the category of Heyting frames and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices…
In this paper we provided an explicit construction for the left adjoint of the forgetful functor from the category of Heyting algebras to that of Hilbert algebras. This functor factorizes through the free implicative semilattice extension of a Hilbert algebra of Celani and Jansana [On the free implicative semilattice extension of a Hilbert algebra}. Mathematical Logic Quarterly 58, 3 (2012), 188--207].
The purpose of this note is two-fold. Firstly, we prove that the variety RDMSH1 of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of RDMSH1. Secondly, we give a concrete description of the lattice of subvarieties of the variety RDQDStSH1 of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains RDMSH1 . Furthermore, we prove that every subvariety of RDQDStSH1, and hence of RDMSH1, has Amalgamation Property. The note concludes with some open problems for further investigation.
Varieties of strictly n-generated Heyting algebras
本文构造无限 $(n+1)$-生成 Heyting 代数 $H_n$，其仅含有限 $n$-生成子代数，从而证明存在相应变种，并否定回答了 $n=2$ 时 Bezhanishvili 与 Grigolia 的问题。
It was shown recently that epimorphisms need not be surjective in a variety K of Heyting algebras, but only one counter-example was exhibited in the literature until now. Here, a continuum of such examples is identified, viz. the variety generated by the Rieger-Nishimura lattice, and all of its (locally finite) subvarieties that contain the original counter-example K. It is known that, whenever a variety of Heyting algebras has finite depth, then it has surjective epimorphisms. In contrast, we show that for every integer n greater or equal than 2, the variety of all Heyting algebras of width a…
On a Generalization of Heyting Algebras I
本文作为系列论文首篇，研究推广Heyting代数的$\nabla$-代数的代数-拓扑性质，刻画其簇的次直不可约元与单元素，证明Dedekind-MacNeille完备化封闭性，给出典范构造与Kripke表示并建立部分簇的合并性质。
Completeness and interpolation for intuitionistic infinitary predicate logic, in connection to finitizing the class of representable Heyting polyadic algebras
研究了Heyting多进代数的各种约简的表示定理。超合并证明了几个（自然约简）和我们的结果进行了比较，在经典代数逻辑处理圆柱和polyadic（布尔代数）的有限性问题。我们还证明了几个新的整齐嵌入定理，并得到了基于（广义）Kripke语义的可表示代数类与具有整齐嵌入性质的代数类，即那些代数是具有$ω$额外维数的代数的次整齐约化.
It is known that exactly eight varieties of Heyting algebras have a model-completion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms have a certain embedding property. For four of these six varieties (those which are locally finite) this actually …
Embeddings into Free Heyting Algebras and Translations into Intuitionistic Propositional Logic
我们发现一个翻译具有特别好的性质，从直觉命题逻辑在可数多个变量的直觉命题逻辑在两个变量。此外，还证明了从任何可数逻辑到二元直觉命题逻辑的可能不那么好的翻译的存在性。证明了从经典逻辑到直觉主义命题逻辑的翻译不存在，它保留了“与”和“或”，但不一定是“真”。这些结果的翻译遵循额外的结果嵌入到自由海廷代数。
We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra which embeds in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regu…
On a Generalization of Heyting Algebras II
该研究作为Heyting代数推广研究的续篇，引入$\nabla$-空间建立相关$\nabla$-代数范畴的对偶理论，给出多类空间上动态拓扑系统的代数刻画与部分$\nabla$-代数的环论表示，还构建了对应逻辑系统的多种语义并证明部分系统的演绎插值定理。
In this paper, we focus on the variety DHMSH of dually hemimorphic semi-Heyting algebras from a logical point of view. Firstly, we present a Hilbert-style axiomatization of a new logic called Dually hemimorphic semi-Heyting logic (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense…
Etale algebras over finite Heyting algebras
该研究引入平展海廷H代数概念，在有限海廷代数情形下建立其范畴论对偶性与公理化恒等式，证明相关范畴等价性并得出平展海廷H代数范畴有限余极限更易计算的结论。
Injectives in the variety generated by a finite subdirectly irreducible Heyting algebra with involution
证明了任何具有对合的有限次直不可约Heyting代数都是拟素的，并且有限次直不可约Heyting代数生成的簇中的内射代数恰是该代数的某个直幂的对角子代数，它们是完备的格.
Codimension and pseudometric in co-Heyting algebras
本文对分配有界格$L$的每个元素引入维数和余维数的概念。证明了当$L$是一个co-Heyting代数时，这些概念具有良好的性质.在这种情况下，余维数产生了一个伪度量的$L$，它满足超度量三角不等式。我们证明了$L$关于这个伪度量的Hausdorff完备化恰好是它的所有有限维子的投影极限。这种完备化具有一些常见的度量性质，例如每个单调序列在紧子集中的收敛性。它与$L$的profinite完备化一致当且仅当它是紧的或等价当$L$的每个有限维商是有限的。在这种情况下，我们说$L$是预紧的。如果$L$是预紧的，并且是Hausdorff的，那么它继承了它的完备化的许多显著性质，特别是那些关于并/交不可约元的性质。由于每一个新提出的co-Heyting代数都是预紧Hausdorff代数，所以我们证明的关于后者代数结构的所有结果特别适用于前者。作为应用，我们得到了对于每个正整数$n,d$，存在一个项$t_{n,d}$，使得在由$n$元组$a$生成的每个余Heyting代数中，$t_{n,d}(a)$恰好是余维数$d$的最大元素.
The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new mathematical family called Heyting structures, which are similar to semirings, but with Heyting-algebra operators in place of the usual arithmetic operators usually seen in semirings. The impact of the zero-sum free property of semirings on Heyting structures is shown as also the condition under which it is possible to extend one Heyting structure to another. It is …
We present a category equivalent to that of semi-Nelson algebras. The objects in this category are pairs consisting of a semi-Heyting algebra and one of its filters. The filters must contain all the dense elements of the semi-Heyting algebra and satisfy an additional technical condition. We also show that in the case of dually hemimorphic semi-Nelson algebras, the filters are not necessary and the category is equivalent to that of dually hemimorphic semi-Heyting algebras.
Let p be prime number, K be a p-adically closed field, X $\subseteq$ K^m a semi-algebraic set defined over K and L(X) the lattice of semi-algebraic subsets of X which are closed in X. We prove that the complete theory of L(X) eliminates the quantifiers in a certain language LASC, the LASC-structure on L(X) being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field. We classify these LASC-structures up to elementary equivalence, and get in particular that the complete theory of L(K^m) only depends on m, not on K nor eve…
Bounded depth in Hilbert algebras
研究hilbert代数中深度有界条件的等价性，扩展heyting代数中类似结果
Finitely generated free Heyting algebras via Birkhoff duality and coalgebra
完全由秩1公理公理化的代数是函子的代数，因此自由代数可以通过直接极限过程得到。对偶地，最终的余代数可以通过逆极限过程得到。为了探索这种方法的限制，我们看看海廷代数具有混合秩0-1公理化。我们将看到，海廷代数是特殊的，因为它们几乎是秩1公理化的，可以处理的秩1 coalgebraic方法略有变化。
Colimits of Heyting Algebras through Esakia Duality
本文通过推广Ghilardi构造自由Heyting代数的方法，研究Heyting代数的余极限及其在Esakia空间中的极限，同时证明Heyting代数的余分配性及 amalgamation性质。
By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of count…
A topos for étale-finite Heyting algebras
针对 Pitts 提出的每个 Heyting 代数是否都是某个初等拓扑斯的真值格问题，本文通过 Esakia 对偶性为每个 étale-有限 Heyting 代数构造了一个初等拓扑斯，其真值格同构于该代数，从而扩展了已知肯定答案的 Heyting 代数类。
An explicit Kuznetsov-Muravitsky enrichment
本文构造任意Heyting代数到Kuznetsov-Muravitsky代数归约的显式嵌入，并代数证明该归约属于原代数生成的Heyting簇。
Connexive logics and connexive semi-Heyting algebras
本文研究了连通逻辑与半海廷代数，定义了新的逻辑系统并探讨了其代数语义及性质。
Regular Double $p$-Algebras: A converse to a Katriňák's Theorem, and Applications
本文证明正则双 $p$-代数中 Katriňák 定理的逆定理，并据此建立多类正则代数簇的项等价关系，推出其子簇格与相关逻辑扩张格均连续统基数。
Coexact completion of profinite Heyting algebras and uniform interpolation
本文将有限生成自由Heyting代数的层表示分解为有限完备化，确定有限Heyting代数的对偶范畴为K-拓扑斯的满子范畴，推广一致内插性质并建立K-拓扑斯的拓扑斯理论性质。
Scott topologies on products of countable complete Heyting algebras
本文证明斯科特拓扑与可数表示框架的任意乘积相容且乘积空间是素谱的，并指出可数生成不足以保证该性质，同时给出阿廷空间与代数 dcpo 的关联及二分法定理。
A dual characterisation of simple and subdirectly-irreducible temporal Heyting algebras
建立了时态海廷代数与时态埃萨基亚空间的对偶性，利用可达性概念给出了简单和次直不可约代数的格论与序拓扑刻画，并证明了时态海廷演算的有限模型性质。
Choice-free Topological Duality for Implicative Lattices and Heyting Algebras
本研究通过证明蕴涵格的无选择公理表示与对偶定理，构建了可同时解释$\boldsymbol{\text{IPC}}$及更弱逻辑的通用语义框架，并将其特化为Heyting代数与拓扑分类标架范畴的无选择公理拓扑对偶。
Distributive lattices in o-minimal structures
研究o-极小结构中可定义的分配格与Heyting代数，给出实闭域扩张上可定义的一维有界分配格的完整描述，并证明可定义的Birkhoff表示定理，用于分类Heyting代数语言中单变量方程在给定代数最大维子集上的可满足性。
Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms between free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly…
Uniform Local Tabularity in Intuitionistic Logic
本文研究直觉主义逻辑中一致局部表格性的强化概念，通过Heyting代数的迭代蕴涵生成，证明n一致局部有限Heyting代数构成簇，并给出n<3时的公理化，同时构造反例回答Shehtman问题。
Tense SHn--algebras
1982年，L. Iturrioz引入了n阶对称Heyting代数（或SHn-代数）。本文定义并研究了时态SHn-代数，即赋两个时态算子的SHn-代数。这些代数构成了紧张Lukasiewicz-Moisil代数的推广。我们的主要兴趣是紧张SHn-代数的对偶理论。为了做到这一点，我们需要Esakia的对偶Heyting代数和Goldblatt的对偶有界分配格与操作。
Coding-Logic Correspondence: Turning Information and Communication Networks into Logical Formulae via Hypergraph Heyting Algebra
通过超图Heyting代数将信息和通信网络转化为逻辑公式，实现编码设置与逻辑公式的对应关系。
Intuitionistic logic, dual intuitionistic logic, and modality
本文通过余海廷代数与拓扑空间的关系研究对偶直觉主义逻辑的语义，并证明其与直觉主义逻辑通过Kripke语义共同关联于模态逻辑S4。
A hyperamalgamation property
提出超融合性质，证明格、半格、布尔代数和对偶海廷代数具有该性质，并应用于带算子的代数结构融合性。
Hennessy-Milner Type Theorems for Fuzzy Multimodal Logics Over Heyting Algebras
本文针对基于完备线性序Heyting代数的模糊多模态逻辑，引入Kripke模型间弱互模拟概念，证明三个Hennessy-Milner型定理，揭示不同公式集下弱互模拟与特定模糊互模拟的一致性。
Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior in terms of Heyting-valued structures. In this paper, we first provide a systematic treatment of sheaves of structures and Heyting-valued structures from the viewpoint of categorical logic. We then prove a form of Łoś's theorem for Heyting-valued structures. We also give a characterization of Heyting-valued structures for which Łoś's theorem holds with respect to any maximal filter.
In [{\it On the free implicative semilattice extension of a Hilbert algebra}. Mathematical Logic Quarterly 58, 3 (2012), 188--207], Celani and Jansana give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in last section we describe a functor from the algebraic category of Hilbert al…
Degree of Satisfiability in Heyting Algebras
本文研究了Heyting代数和直觉逻辑中的可满足度问题，分类了单自由变元方程的有限可满足间隙，证明了非布尔Heyting代数中排中律可满足度不超过2/3，并将结果推广至无限代数及拓扑、黑盒代数与逻辑哲学应用。
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