Results for 'Subdirectly irreducible'

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  1.  17
    Subdirectly Irreducible Separable Dynamic Algebras.Sandra Marques Pinto & M. Teresa Oliveira-Martins - 2010 - Mathematical Logic Quarterly 56 (4):442-448.
    A characterization of the subdirectly irreducible separable dynamic algebras is presented. The notions develo- ped for this study were also suitable to describe the previously found class of simple separable dynamic algebras.
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  2.  11
    Subdirectly Irreducible State-Morphism BL-Algebras.Anatolij Dvurečenskij - 2011 - Archive for Mathematical Logic 50 (1-2):145-160.
    Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201–206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138–152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic (...)
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  3.  7
    Subdirectly Irreducible IKt-Algebras.Aldo V. Figallo, Inés Pascual & Gustavo Pelaitay - 2017 - Studia Logica 105 (4):673-701.
    The IKt-algebras that we investigate in this paper were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. Now we characterize by topological methods the subdirectly irreducible IKt-algebras and particularly the simple IKt-algebras. Finally, we consider the particular cases of finite IKt-algebras and complete IKt-algebras.
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  4.  38
    A Dual Characterization of Subdirectly Irreducible BAOs.Yde Venema - 2004 - Studia Logica 77 (1):105 - 115.
    We give a characterization of the simple, and of the subdirectly irreducible boolean algebras with operators (including modal algebras), in terms of the dual descriptive frame, or, topological relational structure. These characterizations involve a special binary topo-reachability relation on the dual structure; we call a point u a topo-root of the dual structure if every ultrafilter is topo-reachable from u. We prove that a boolean algebra with operators is simple iff every point in the dual structure is a (...)
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  5.  22
    Subdirectly Irreducible Modal Algebras and Initial Frames.Sambin Giovanni - 1999 - Studia Logica 62 (2):269-282.
    The duality between general frames and modal algebras allows to transfer a problem about the relational (Kripke) semantics into algebraic terms, and conversely. We here deal with the conjecture: the modal algebra A is subdirectly irreducible (s.i.) if and only if the dual frame A* is generated. We show that it is false in general, and that it becomes true under some mild assumptions, which include the finite case and the case of K4. We also prove that a (...)
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  6.  24
    On Subdirectly Irreducible OMAs.Richard Holzer - 2004 - Studia Logica 78 (1-2):261 - 277.
    In this paper some properties of epi-representations and Schmidt-congruence relations of orthomodular partial algebras are investigated and an infinite list of OMA-epi-subdirectly irreducible orthomodular partial algebras will be constructed.
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  7.  38
    On Ockham Algebras: Congruence Lattices and Subdirectly Irreducible Algebras.P. Garcia & F. Esteva - 1995 - Studia Logica 55 (2):319 - 346.
    Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results (...)
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  8.  39
    Subdirectly Irreducible Residuated Semilattices and Positive Universal Classes.Jeffrey S. Olson - 2006 - Studia Logica 83 (1-3):393-406.
    CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)k ≤ (x ⋀ e)k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k).
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  9.  13
    Subdirectly Irreducible P-Compatible Abelian Groups.Krystyna Mruczek - 2003 - Bulletin of the Section of Logic 32 (1/2):57-63.
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  10. Subdirectly Irreducible Separable Dynamic Algebras.Sandra Marques Pinto & M. Teresa F. Oliveira Martins - 2010 - Mathematical Logic Quarterly 56 (4):442-448.
     
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  11.  10
    Quasi‐Stone Algebras.Nalinaxi H. Sankappanavar & Hanamantagouda P. Sankappanavar - 1993 - Mathematical Logic Quarterly 39 (1):255-268.
    The purpose of this paper is to define and investigate the new class of quasi-Stone algebras . Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an -chain. MSC: 03G25, 06D16, 06E15.
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  12.  19
    Hilbert Algebras with a Modal Operator $${\Diamond}$$ ◊.Sergio A. Celani & Daniela Montangie - 2015 - Studia Logica 103 (3):639-662.
    A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie . In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊ . We give a topological representation for these algebras using the topological spectral-like representation for Hilbert algebras with supremum given in (...)
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  13.  20
    De Morgan Algebras with a Quasi-Stone Operator.T. S. Blyth, Jie Fang & Lei-bo Wang - 2015 - Studia Logica 103 (1):75-90.
    We investigate the class of those algebras in which is a de Morgan algebra, is a quasi-Stone algebra, and the operations \ and \ are linked by the identity x**º = x*º*. We show that such an algebra is subdirectly irreducible if and only if its congruence lattice is either a 2-element chain or a 3-element chain. In particular, there are precisely eight non-isomorphic subdirectly irreducible Stone de Morgan algebras.
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  14.  8
    Weak‐Quasi‐Stone Algebras.Sergio A. Celani & Leonardo M. Cabrer - 2009 - Mathematical Logic Quarterly 55 (3):288-298.
    In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and (...) irreducible algebras. (shrink)
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  15.  27
    Expansions of Semi-Heyting Algebras I: Discriminator Varieties.H. P. Sankappanavar - 2011 - Studia Logica 98 (1-2):27-81.
    This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion (...)
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  16.  4
    Locally Finite Reducts of Heyting Algebras and Canonical Formulas.Guram Bezhanishvili & Nick Bezhanishvili - 2017 - Notre Dame Journal of Formal Logic 58 (1):21-45.
    The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting algebras give rise to the (...)
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  17.  34
    On the Representation of N4-Lattices.Sergei P. Odintsov - 2004 - Studia Logica 76 (3):385 - 405.
    N4-lattices provide algebraic semantics for the logic N4, the paraconsistent variant of Nelson's logic with strong negation. We obtain the representation of N4-lattices showing that the structure of an arbitrary N4-lattice is completely determined by a suitable implicative lattice with distinguished filter and ideal. We introduce also special filters on N4-lattices and prove that special filters are exactly kernels of homomorphisms. Criteria of embeddability and to be a homomorphic image are obtained for N4-lattices in terms of the above mentioned representation. (...)
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  18.  22
    Basic Hoops: An Algebraic Study of Continuous T -Norms.P. Aglianò, I. M. A. Ferreirim & F. Montagna - 2007 - Studia Logica 87 (1):73 - 98.
    A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that ([ 0,1], *, 1) is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and ([ 0,1], *, →, 1) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras ([ 0,1], *, →, 1), where (...)
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  19.  12
    A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic.Carles Noguera & Tomáš Lávička - 2017 - Studia Logica 105 (3):521-551.
    In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively subdirectly irreducible models. We identify two syntactical notions formulated in terms of intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, (...)
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  20.  22
    Equational Bases for Joins of Residuated-Lattice Varieties.Nikolaos Galatos - 2004 - Studia Logica 76 (2):227 - 240.
    Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL + (...)
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  21.  11
    Basic Hoops: An Algebraic Study of Continuous T-Norms.P. Aglianò, I. M. A. Ferreirim & F. Montagna - 2007 - Studia Logica 87 (1):73-98.
    A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the structure of the (...)
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  22.  29
    Distributive Lattices with a Dual Homomorphic Operation.Alasdair Urquhart - 1979 - Studia Logica 38 (2):201 - 209.
    The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.
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  23.  33
    A Model-Theoretic Analysis of Fidel-Structures for mbC.Marcelo E. Coniglio - forthcoming - In Can Baskent and Thomas Ferguson (ed.), Graham Priest on Dialetheism and Paraconsistency. Springer.
    In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to (...)
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  24.  15
    On N -Contractive Fuzzy Logics.Rostislav Horčík, Carles Noguera & Milan Petrík - 2007 - Mathematical Logic Quarterly 53 (3):268-288.
    It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation (...)
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  25.  79
    Quasivarieties with Definable Relative Principal Subcongruences.Anvar M. Nurakunov & M. M. Stronkowski - 2009 - Studia Logica 92 (1):109-120.
    For quasivarieties of algebras, we consider the property of having definable relative principal subcongruences, a generalization of the concepts of definable relative principal congruences and definable principal subcongruences. We prove that a quasivariety of algebras with definable relative principal subcongruences has a finite quasiequational basis if and only if the class of its relative subdirectly irreducible algebras is strictly elementary. Since a finitely generated relatively congruence-distributive quasivariety has definable relative principal subcongruences, we get a new proof of the (...)
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  26.  18
    State-Morphism MV-Algebras.Antonio Di Nola & Anatolij Dvurečenskij - 2009 - Annals of Pure and Applied Logic 161 (2):161-173.
    We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such state MV-algebras with the category of unital Abelian ℓ-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.
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  27.  28
    An Intriguing Logic with Two Implicational Connectives.Lloyd Humberstone - 2000 - Notre Dame Journal of Formal Logic 41 (1):1-40.
    Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between (...)
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  28.  62
    Ockham Algebras with Balanced Double Pseudocomplementation.Jie Fang - 2008 - Studia Logica 90 (2):189-209.
    In this paper, we introduce a variety bdO of Ockham algebras with balanced double pseudocomplementation, consisting of those algebras of type where is an Ockham algebra, is a double p -algebra, and the operations and are linked by the identities [ f ( x )]* = [ f ( x )] + = f 2 ( x ), f ( x *) = x ** and f ( x + ) = x ++ . We give a description of the (...)
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  29.  29
    Distributive Lattices with an Operator.Alejandro Petrovich - 1996 - Studia Logica 56 (1-2):205 - 224.
    It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly (...) algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences. (shrink)
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  30.  25
    Flat Algebras and the Translation of Universal Horn Logic to Equational Logic.Marcel Jackson - 2008 - Journal of Symbolic Logic 73 (1):90-128.
    We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is (...)
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  31.  43
    The Variety of Lattice Effect Algebras Generated by MV-Algebras and the Horizontal Sum of Two 3-Element Chains.Radomír Halaš - 2008 - Studia Logica 89 (1):19-35.
    It has been recently shown [4] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The open problem whether all subdirectly irreducible distributive lattice effect algebras are just subdirectly irreducible MV-chains and the horizontal sum of two 3-element chains is in the paper transferred into a more tractable one. We prove that modulo distributive lattice effect algebras, the variety generated by MV-algebras and is definable by three simple (...)
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  32.  43
    Optimal Natural Dualities for Varieties of Heyting Algebras.B. A. Davey & H. A. Priestley - 1996 - Studia Logica 56 (1-2):67 - 96.
    The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.
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  33.  23
    Semi-Demorgan Algebras.David Hobby - 1996 - Studia Logica 56 (1-2):151 - 183.
    Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of partial diagrams is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest (...)
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  34.  25
    Birkhoff-Like Sheaf Representation for Varieties of Lattice Expansions.Hector Gramaglia & Diego Vaggione - 1996 - Studia Logica 56 (1-2):111 - 131.
    Given a variety we study the existence of a class such that S1 every A can be represented as a global subdirect product with factors in and S2 every non-trivial A is globally indecomposable. We show that the following varieties (and its subvarieties) have a class satisfying properties S1 and S2: p-algebras, distributive double p-algebras of a finite range, semisimple varieties of lattice expansions such that the simple members form a universal class (bounded distributive lattices, De Morgan algebras, etc) and (...)
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  35.  56
    Varieties of Three-Valued Heyting Algebras with a Quantifier.M. Abad, J. P. Díaz Varela, L. A. Rueda & A. M. Suardíaz - 2000 - Studia Logica 65 (2):181-198.
    This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q 3 and we construct the lattice (...)
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  36.  13
    Even More About the Lattice of Tense Logics.Marcus Kracht - 1992 - Archive for Mathematical Logic 31 (4):243-257.
    The present paper is based on [11], where a number of conjectures are made concerning the structure of the lattice of normal extensions of the tense logicKt. That paper was mainly dealing with splittings of and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of will be the splitting theorem of [8]. In [11] it was conjectured that each finite subdirectly irreducible algebra splits the lattice of (...)
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  37.  20
    Pseudomonadic Algebras as Algebraic Models of Doxastic Modal Logic.Nick Bezhanishvili - 2002 - Mathematical Logic Quarterly 48 (4):624-636.
    We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; Ordertopological representation of pseudomonadic algebras; Complete description of the lattice of subvarieties of the variety of (...)
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  38. Varieties of Three-Values Heyting Algebras with a Quantifier.Manuel Abad, J. P. Diaz Varela & L. A. Rueda - 2000 - Studia Logica 65 (2):181-198.
    This paper is devoted to the study of some subvarieties of the variety Q of Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q subscript 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Q is far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q subscript 3 and (...)
     
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  39.  24
    Algebras and Matrices for Annotated Logics.R. A. Lewin, I. F. Mikenberg & M. G. Schwarze - 2000 - Studia Logica 65 (1):137-153.
    We study the matrices, reduced matrices and algebras associated to the systems SAT of structural annotated logics. In previous papers, these systems were proven algebraizable in the finitary case and the class of matrices analyzed here was proven to be a matrix semantics for them.We prove that the equivalent algebraic semantics associated with the systems SAT are proper quasivarieties, we describe the reduced matrices, the subdirectly irreducible algebras and we give a general decomposition theorem. As a consequence we (...)
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  40.  13
    The Syntax and Semantics of Entailment in Duality Theory.B. A. Davey, M. Haviar & H. A. Priestley - 1995 - Journal of Symbolic Logic 60 (4):1087-1114.
    Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...)
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  41.  15
    De Morgan Heyting Algebras Satisfying the Identity Xn() X(N+1)().Valeria Castaño & Marcela Muñoz Santis - 2011 - Mathematical Logic Quarterly 57 (3):236-245.
    In this paper we investigate the sequence of subvarieties equation imageof De Morgan Heyting algebras characterized by the identity xn ≈ x. We obtain necessary and sufficient conditions for a De Morgan Heyting algebra to be in equation image by means of its space of prime filters, and we characterize subdirectly irreducible and simple algebras in equation image. We extend these results for finite algebras in the general case equation image. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, (...)
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  42.  14
    Weak-Quasi-Stone Algebras.Sergio A. Celani & Leonardo M. Cabrer - 2009 - Mathematical Logic Quarterly 55 (3):288-298.
    In this paper we shall introduce the variety WQS of weak-quasi-Stone algebras as a generalization of the variety QS of quasi-Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬-lattices to give a duality for WQS. We prove that a weak-quasi-Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and (...) irreducible algebras. (shrink)
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  43.  11
    Linear Heyting Algebras with a Quantifier.Laura Rueda - 2001 - Annals of Pure and Applied Logic 108 (1-3):327-343.
    A Q -Heyting algebra is an algebra of type such that is a Heyting algebra and the unary operation ∇ satisfies the conditions ∇0=0, a ∧∇ a = a , ∇=∇ a ∧∇ b and ∇=∇ a ∨∇ b , for any a , b ∈ H . This paper is devoted to the study of the subvariety QH L of linear Q -Heyting algebras. Using Priestley duality we investigate the subdirectly irreducible linear Q -Heyting algebras and, as (...)
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  44.  22
    Existentially Closed Algebras and Boolean Products.Herbert H. J. Riedel - 1988 - Journal of Symbolic Logic 53 (2):571-596.
    A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γ a (K) has the Fraser-Horn property. If $\lbrack a \neq b\rbrack \cap \lbrack c \neq d\rbrack = \varnothing$ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is (...)
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  45.  18
    Fuzzy Propositional Logic. Algebraic Approach.Slava Meskhi - 1977 - Studia Logica 36 (3):189 - 194.
    The present paper contains some technical results on a many-valued logic with truth values from the interval of real numbers [0; 1]. This logic, discussed originally in [1], latter in [2] and [3], was called the logic of fuzzy concepts. Our aim is to give an algebraic axiomatics for fuzzy propositional logic. For this purpose the variety of L-algebras with signature en- riched with a unary operation { involution is stud- ied. A one-to-one correspondence between congruences on an LI-algebra and (...)
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  46.  13
    Characteristic Formulas of Partial Heyting Algebras.Alex Citkin - 2013 - Logica Universalis 7 (2):167-193.
    The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting (...)
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  47.  7
    Reductio Ad Contradictionem: An Algebraic Perspective.Adam Přenosil - 2016 - Studia Logica 104 (3):389-415.
    We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in (...)
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  48.  16
    An Infinitary Extension of Jankov’s Theorem.Yoshihito Tanaka - 2007 - Studia Logica 86 (1):111 - 131.
    It is known that for any subdirectly irreducible finite Heyting algebra A and any Heyting algebra B, A is embeddable into a quotient algebra of B, if and only if Jankov’s formula χ A for A is refuted in B. In this paper, we present an infinitary extension of the above theorem given by Jankov. More precisely, for any cardinal number κ, we present Jankov’s theorem for homomorphisms preserving infinite meets and joins, a class of subdirectly (...) complete κ-Heyting algebras and κ-infinitary logic, where a κ-Heyting algebra is a Heyting algebra A with # ≥ κ and κ-infinitary logic is the infinitary logic such that for any set Θ of formulas with # Θ ≥ κ, ∨Θ and ∧Θ are well defined formulas. (shrink)
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  49.  7
    Heyting Algebras with Operators.Yasusi Hasimoto - 2001 - Mathematical Logic Quarterly 47 (2):187-196.
    In this paper, we will give a general description of subdirectly irreducible Heyting algebras with operators under some weak conditions, which includes the finite case, the normal case and the case for Boolean algebras with diamond operator. This can be done by normalizing these operators. This answers the question posed in Wolter [4].
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  50.  13
    On Endomorphisms of Ockham Algebras with Pseudocomplementation.T. S. Blyth & J. Fang - 2011 - Studia Logica 98 (1-2):237-250.
    A pO -algebra $${(L; f, \, ^{\star})}$$ is an algebra in which ( L ; f ) is an Ockham algebra, $${(L; \, ^{\star})}$$ is a p -algebra, and the unary operations f and $${^{\star}}$$ commute. Here we consider the endomorphism monoid of such an algebra. If $${(L; f, \, ^{\star})}$$ is a subdirectly irreducible pK 1,1 - algebra then every endomorphism $${\vartheta}$$ is a monomorphism or $${\vartheta^3 = \vartheta}$$ . When L is finite the endomorphism monoid of (...)
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