Residuated logics based on strict triangular norms with an involutive negation

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t‐norms and their dual t‐conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t‐norms which are different from the product logic. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The finite model property for semilinear substructural logics

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.     

A Gentzen system for involutive residuated lattices

We establish a cut-free Gentzen system for involutive residuated lattices and provide an algebraic proof of completeness. As a result we conclude that the equational theory of involutive residuated lattices is decidable. The connection to noncommutative linear logic is outlined. Received July 22, 2004; accepted in final form July 19, 2005.     

On involutive FL e -monoids

The paper deals with involutive FL _e -monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FL _e -monoids over lattices are exactly involutive FL _e -algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FL _e -monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FL _e -chains are classified by using the notion of rank of involutive FL _e -chains, and a kind of duality is developed between positive and non-positive rank algebras. As a side effect, it is shown that the substructural logic IUL plus t ? f does not have the finite model property.     

Method for Solving the Multidimensional n‐Wave Resonant Equations and Geometry of Generalized Darboux‐Manakov‐Zakharov Systems

The intrinsic geometric properties of generalized Darboux‐Manakov‐Zakharov systems of semilinear partial differential equations (1) for a real‐valued function u(x₁, …, x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1) , essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n‐wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n‐wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.     

Distinguishing standard SBL‐algebras with involutive negations by propositional formulas

Propositional fuzzy logics given by a combination of a continuous SBL t‐norm with finitely many idempotents and of an involutive negation are investigated. A characterization of continuous t‐norms which, in combination with different involutive negations, yield either isomorphic algebras or algebras with distinct and incomparable sets of propositional tautologies is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong negation in intuitionistic style sequent systems for residuated lattices

We study the sequent system mentioned in the author's work 18 as CyInFL with ‘intuitionistic’ sequents. We explore the connection between this system and symmetric constructive logic of Zaslavsky 40 and develop an algebraic semantics for both of them. In contrast to the previous work, we prove the strong completeness theorem for CyInFL with ‘intuitionistic’ sequents and all of its basic variants, including variants with contraction. We also show how the defined classes of structures are related to cyclic involutive FL‐algebras and Nelson FL_ew‐algebras. In particular, we prove the definitional equivalence of symmetric constructive FL_ewc‐algebras (algebraic models of symmetric constructive logic) and Nelson FL_ew‐algebras (algebras introduced by Spinks and Veroff 33 , 34 as the termwise equivalent definition of Nelson algebras). Because of the strong completeness theorem that covers all basic variants of CyInFL with ‘intuitionistic’ sequents, we rename this sequent system to symmetric constructive full Lambek calculus (). We verify the decidability of this system and its basic variants, as we did in the case of their distributive cousins 18 . As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL‐algebras are decidable.     

Join-semilattices whose sections are residuated PO-monoids

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras. This work was supported by the Czech Government via the project MSM6198959214.     

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant , namely is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that is an involutive negation. However, for t-norms without non-trivial zero divisors, is G?del negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. Received: 14 April 1998     

Involutive invariants of the second kind

Let S be a Krull domain which is divisorially étale of degree two over a Krull domain R. The main purpose of this text is to introduce and study involutive invariants of the second kind of S with respect to R, as well as developing techniques allowing to calculate these in terms of other, classical and involutive, invariants.     

Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.     

Recursive logic frames

We define the concept of a logic frame , which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete (recursively compact , ?₀‐compact ), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least λ ” completeness always implies .0‐compactness. On the other hand we show that a recursively compact logic frame need not be ?₀‐compact. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bisimulations between generalized Veltman models and Veltman models

Interpretability logic is an extension of provability logic. Veltman models and generalized Veltman models are two semantics for interpretability logic. We consider a connection between Veltman semantics and generalized Veltman semantics. We prove that for a complete image‐finite generalized Veltman modelW there is a Veltman model W ′ that is bisimular to W. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Glivenko theorems and negative translations in substructural predicate logics

Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFL _e . It is shown that there exists the weakest logic over QFL _e among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied by using the same approach. First, a negative translation, called extended Kuroda translation is introduced. Then a translation result of an arbitrary involutive substructural predicate logics over QFL _e is shown, and the existence of the weakest logic is proved among such logics for which the extended Kuroda translation works. They are obtained by a slight modification of the proof of the Glivenko theorem. Relations of our extended Kuroda translation with other standard negative translations will be discussed. Lastly, algebraic aspects of these results will be mentioned briefly. In this way, a clear and comprehensive understanding of Glivenko theorems and negative translations will be obtained from a substructural viewpoint.     

Quantum logics with the Riesz Interpolation Property

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Implementation for the Algorithm of Janet bases of Linear Differential Ideals in the Maple System

In this paper, an algorithm for computing the Janet bases of linear differential equations is described, which is the differential analogue of the algorithm JanetBasis improved by Gerdt. An implementation of the algorithm in Maple is given. The implemented algorithm includes some subalgorithms: Janet division,Pommaret division, the judgement of involutive divisor and reducible, the judgement of conventional divisor and reducible, involutive normal form and conventional normal form, involutive autoreduction and conventional autoreduction, PJ-autoreduction and so on. As an application, the Janet Bases of the determining system of classical Lie symmetries of some partial differential equations are obtained using our package.     

Finitely stable racks and rack representations

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the strong representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the Pontryagin dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.     

On linear combinations of two tripotent, idempotent, and involutive matrices

Let A=c₁A₁+c₂A₂, wherec₁, c₂ are nonzero complex numbers and (A₁,A₂) is a pair of two n×n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A=c₁A₁+c₂A₂ is (i) a tripotent or an involutive matrix when are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when are involutive matrices, and (iii) an involutive matrix when are involutive or idempotent matrices.