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In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored. [ABSTRACT FROM AUTHOR]

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic.

In this paper, we study three representations of lattices by means of a set with a binary relation of compatibility in the tradition of Ploscica. The standard representations of complete ortholattices and complete perfect Heyting algebras drop out as special cases of the first representation, while the second covers arbitrary complete lattices, as well as complete lattices equipped with a negation we call a protocomplementation. The third topological representation is a variant of that of Craig, Haviar, and Priestley. We then extend each of the three representations to lattices with a multiplicative unary modality; the representing structures, like so-called graph-based frames, add a second relation of accessibility interacting with compatibility. The three representations generalize possibility semantics for classical modal logics to non-classical modal logics, motivated by a recent application of modal orthologic to natural language semantics.

In this paper, we study α-fuzzy ideals in C-algebras whose truth degrees are in a complete Heyting algebra. We provide several characterizations for fuzzy ideals to be an α-fuzzy ideal. One of the key contributions of this manuscript is the investigation of the smallest α-fuzzy ideal containing a given fuzzy ideal. By establishing the existence and properties of this smallest α-fuzzy ideal, we shed light on the structure and behavior of α-fuzzy ideals in C-algebras. Furthermore, we prove that the class of α-fuzzy ideals forms a complete lattice. We obtain the closure operator on the class of fuzzy ideals FI(A), where the closed elements correspond to the α-fuzzy ideals. We explore the conditions under which every fuzzy ideal in a given C-algebra becomes an α-fuzzy ideal. Finally, we study the space of prime α-fuzzy ideals in C-algebras and we derive a necessary and sufficient condition for this space to be a T1 space. [ABSTRACT FROM AUTHOR]

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety DH of all DH-algebras is congruence distributive. The lattice of all subvarieties of DH is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed. [ABSTRACT FROM AUTHOR]

We show that intuitionistic propositional logic is Carnap categorical: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds with respect to the most well-known semantics relative to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, topological semantics, and algebraic semantics. These facts turn out to be consequences of an observation about interpretations in Heyting algebras. [ABSTRACT FROM AUTHOR]

It is well known that the lattice Id c G of all principal ℓ -ideals of any Abelian ℓ -group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some Id c G , via a counterexample of cardinality ℵ 2 . We prove that every completely normal distributive 0-lattice with at most ℵ 1 elements is a homomorphic image of some Id c G . By Stone duality, this means that every completely normal generalized spectral space with at most ℵ 1 compact open sets is homeomorphic to a spectral subspace of the ℓ -spectrum of some Abelian ℓ -group. [ABSTRACT FROM AUTHOR]

ntly, Gautama algebras were defined and investigated as a common generalization of the variety RDBLSt of regular double Stone algebras and the variety RKLSt of regular Kleene Stone algebras, both of which are, in turn, generalizations of Boolean algebras. Those algebras were named in honor and memory of the two founders of Indian Logic{Akshapada Gautama and Medhatithi Gautama. The purpose of this paper is to define and investigate a generalization of Gautama algebras, called \Almost Gautama algebras (AG, for short)." More precisely, we give an explicit description of subdirectly irreducible Almost Gautama algebras. As consequences, explicit description of the lattice of subvarieties of AG and the equational bases for all its subvarieties are given. It is also shown that the variety AG is a discriminator variety. Next, we consider logicizing AG; but the variety AG lacks an implication operation. We, therefore, introduce another variety of algebras called \Almost Gautama Heyting algebras" (AGH, for short) and show that the variety AGH is term-equivalent to that of AG. Next, a propositional logic, called AG (or AGH), is defined and shown to be algebraizable (in the sense of Blok and Pigozzi) with the variety AG, via AGH; as its equivalent algebraic semantics (up to term equivalence). All axiomatic extensions of the logic AG, corresponding to all the subvarieties of AG are given. They include the axiomatic extensions RDBLSt, RKLSt and G of the logic AG corresponding to the varieties RDBLSt, RKLSt, and G (of Gautama algebras), respectively. It is also deduced that none of the axiomatic extensions of AG has the Disjunction Property. Finally, We revisit the classical logic with strong negation CN and classical Nelson algebras CN introduced by Vakarelov in 1977 and improve his results by showing that CN is algebraizable with CN as its algebraic semantics and that the logics RKLSt, RKLStH, 3-valued Lukasivicz logic and the classical logic with strong negation are all equivalent [ABSTRACT FROM AUTHOR]

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The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbertstyle 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 of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi- Heyting algebras, as well as a new axiomatization for the 3-valued Lukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. [ABSTRACT FROM AUTHOR]

We continue our research plan of developing the theory of small and locally small spaces, proposing this theory as a realisation of Grothendieck's idea of tame topology on the level of general topology. In this paper, we develop the theory of Heyting small spaces and prove a new version of Esakia Duality for such spaces. To do this, we notice that spectral spaces may be seen as sober small spaces with all smops compact and introduce the method of the standard spectralification. This helps to understand open continuous definable mappings between definable spaces over o-minimal structures. [ABSTRACT FROM AUTHOR]

In this paper, we defined the notions of (∈, ∈)-fuzzy implicative deductive systems and (∈, ∈ ∨ q)-fuzzy implicative deductive systems of hoops and studied some traits and tried to define some definitions that are equivalent to them. Thus by using the notion of (∈, ∈)-fuzzy deductive system of hoop, we defined a new congruence relation on hoop and show that the algebraic structure that is made by it is a Brouwerian semilattice, Heyting algebra and Wajesberg hoop. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a (strong) hyper RL-ideal in hyper residuated lattices and give some properties and characterizations of them. Next, we characterize the (strong) hyper RL-ideals generated by a subset and give some characterizations of the lattice of these hyper RL-ideals. Particularly, we prove that this lattice is algebraic and compact elements are finitely generated hyper RL-ideals, and obtain some isomorphism theorems. Finally, we introduce the notion of nodal hyper RL-ideals in a hyper residuated lattice and investigate their properties. We prove that the set of nodal hyper RL-ideals is a complete Brouwerian lattice and under suitable operations is a Heyting algebra. [ABSTRACT FROM AUTHOR]

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).

Using the belongs to relation (∍) and quasi-coincident with relation (q) between fuzzy points and fuzzy sets, the notions of an (∊, &#8714)-fuzzy sub-hoop, an (∊, ∊v q)-fuzzy sub-hoop and a (q, ∊v q)-fuzzy sub-hoop are introduced, and several properties are investigated. Characterizations of an (∊, ∊)-fuzzy sub-hoop and an (∊,∊ vq)-fuzzy sub-hoop are displayed. Relations between an (∊,∊)-fuzzy sub-hoop, an (∊,∊ vq)-fuzzy sub-hoop and a (q,∊ vq)-fuzzy sub-hoop are discussed. Conditions for a fuzzy set to be a (q,∊ vq)-fuzzy sub-hoop are considered, and condition for an (∊,∊ vq)-fuzzy sub-hoop to be a (q,∊ vq)-fuzzy sub-hoop are provided. [ABSTRACT FROM AUTHOR]

This paper introduces a notion of intuitionistic topological system. Properties of the proposed system is studied in details. Categorical interrelationships among Heyting algebra, Gödel algebra, Esakia space and proposed intuitionistic topological systems have also been studied. A flavour of Kripke model is given. [ABSTRACT FROM AUTHOR]

In this paper, we intoduce the notion of derivations of state residuated lattices (L,Τ) and discuss some properties of them. We study the related properties of strong derivations and regular derivations of state residuated lattices (L,Τ). Moreover, we propose the notion of (strong) state-morphism residuated lattices and discuss some properties of them. Also, the principal ideal derivation is given and the adjoint of principal ideal derivation is obtained by a Galois connection, and we prove that the set of all principal ideal derivations on state-morphism residuated lattice (L,Τ) can form a bounded distributive lattice. Further, a special kind of set Ima(d,Τ)(L) of a derivation on state residuated lattices (L,Τ) is introduced and we get that Ima(d,Τ)(L) is a lattice ideal of L, when derivation d is a regular ideal derivation. In particular, if L is a linearly ordered residuated lattice, then Ima(d,Τ)(L) is a prime lattice ideal of L. Finally, by using the set Ima(d,Τ)(L) of principal ideal derivations, we give a characterization of a Heyting algebra. [ABSTRACT FROM AUTHOR]

In this paper, we focus on lattice structures of the set of monadic filters (monadic filters, stable monadic filters, involutory monadic filters) of monadic BL-algebras and prove that (a) the classes of all monadic filters in monadic BL-algebras forms a complete Heyting algebras with respect to inclusion; (b) the class of all stable monadic filters relative a monadic filter F in monadic BL-algebras is a complete Boolean algebra with respect to inclusion; (c) the class of all involutory monadic filters relative a monadic filter F in monadic BL-algebras is a complete Boolean algebra with respect to inclusion. These results also provide the solid foundation to study the variety of monadic BL-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ʘ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it. [ABSTRACT FROM AUTHOR]

This paper deals with two constructions in topos theory: po-groups and hypergroups. After a deep analysis of these, we restrict our attention to find a hypergroup associated to a po-group G in a topos E. The method that we use here is based on the Mitchell-B'enabou language. Then, we show that on the negative and positive cones of a po-group G in E, the left and right translations are hyperhomomorphisms in E. Our aim is to find two faithful and left exact functors from the category of po-groups in E to the (smallest in some sense) finitely complete category containing hypergroups in E. A version of this result is also presented on the category of lattices in E instead of po-groups. This version recovers filters and ideals of a lattice in E by means of hyperoperations. We will finish the manuscript by transforming the Heyting algebra structure of the subobject classifier Ω of E to a hypergroup in E. [ABSTRACT FROM AUTHOR]

Let A be an ordered alphabet, A ∗ be the free monoid over A ordered by the Higman ordering, and let F (A ∗) be the set of final segments of A ∗. With the operation of concatenation, this set is a monoid. We show that the submonoid F ∘ (A ∗) ≔ F (A ∗) ∖ { 0̸ } is free. The MacNeille completion N (A ∗) of A ∗ is a submonoid of F (A ∗). As a corollary, we obtain that the monoid N ∘ (A ∗) ≔ N (A ∗) ∖ { 0̸ } is free. We give an interpretation of the freeness of F ∘ (A ∗) in the category of metric spaces over the Heyting algebra V ≔ F (A ∗) , with the non-expansive mappings as morphisms. Each final segment F of A ∗ yields the injective envelope S F of a two-element metric space over V. The uniqueness of the decomposition of F is due to the uniqueness of the block decomposition of the graph G F associated to this injective envelope. [ABSTRACT FROM AUTHOR]

An algebra A = (A, V, Λ, →S-, 0,1} is a semi-Heyting algebra if (A, V, Λ, 0,1) is a bounded lattice, and it satisfies the identities : x Λ (x → y) ≈ x Λ y, x Λ (y → z) ≈ X Λ [(x Λ y), and x → x ≈ 1. SH denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras. They share several important properties with Heyting algebras. An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of S'H is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety SH of aso-ciative type of length 3. Our main result shows that there are 3 such subvarities of SH. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of dual skew Heyting almost distributive lattices (dual skew HADLs) and characterise it in terms of dual HADL. We define an equivalence relation θ on a dual skew HADL L and prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence class is a maximal rectangular subalgebra and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. Moreover, we show that if the set PI(L) of all the principal ideals of an ADL L with 0 is a dual skew Heyting algebra then L becomes a dual skew HADL. Further we present different conditions on which an ADL with 0 becomes a dual skew HADL. [ABSTRACT FROM AUTHOR]

Abstract Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer. [ABSTRACT FROM AUTHOR]

We introduce the notion of a majority category | the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that parallels how Mal'tsev categories capture properties of the category of groups. Among algebraic majority categories are the categories of lattices, Boolean algebras and Heyting algebras. Many geometric categories such as the category of topological spaces, metric spaces, ordered sets, any topos, etc., are comajority categories (i.e. their duals are majority categories), and we show that, under mild assumptions, the only categories which are both majority and comajority, are the preorders. Mal'tsev majority categories provide an alternative generalization of arithmetical categories to protoarithmetical categories in the sense of Bourn. We show that every Mal'tsev majority category is protoarithmetical, provide a counter-example for the converse implication, and show that in the Barr-exact context, the converse implication also holds. We can then conclude that a category is arithmetical if and only if it is a Barr-exact Mal'tsev majority category, recovering in the varietal context a well known result of Pixley. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a logic that corresponds to abstract hoop twist-structures and present some results on this logic. We prove the local deductive theorem for this logic and show that this logic is algebraizable with respect to the quasi-variety of abstract hoop twist-structures. [ABSTRACT FROM AUTHOR]

We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, self-contained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras. [ABSTRACT FROM AUTHOR]

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R n with n ⩾ 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P ⊆ R n , prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P , and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaśkowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds? [ABSTRACT FROM AUTHOR]

We provide simple algebraic proofs of two important facts, due to Zakharyaschev and Esakia, about Grzegorczyk algebras. [ABSTRACT FROM AUTHOR]

The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called “synchronized intervals” introduced by Préville-Ratelle and the present author. Another example is the class of β ( 1 , 0 ) -trees, which has a mysterious involution h proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which is closed under map duality. In this article, we show that we can identify the notions of duality in these three classes using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013). [ABSTRACT FROM AUTHOR]

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, using our earlier results published in 2014, 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. [ABSTRACT FROM AUTHOR]

It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Gödel algebras, relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural logics. On the other hand, it is shown that epimorphisms need not be surjective in a locally finite variety of Heyting or Brouwerian algebras of width 2. It follows that the infinite Beth property is strictly stronger than the so-called finite Beth property, confirming a conjecture of Blok and Hoogland. [ABSTRACT FROM AUTHOR]