In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. (...) Finally, we analyze the relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties k>0 defined in Cignoli and Torrens. (shrink)

In this paper, we show that all semisimple varieties of bounded weak-commutative residuated lattices with an S4-like modal operator are discriminator varieties. We also give a characterization of discriminator and EDPC varieties of bounded weak-commutative residuated lattices with an S4-like modal operator follows.

We prove that all semisimple varieties of FL ew-algebras are discriminator varieties. A characterisation of discriminator and EDPC varieties of FL ew-algebras follows. It matches exactly a natural classification of logics over FL ew proposed by H. Ono.

This work extend to residuated lattices the results of [ 7 ]. It also provides a possible generalization to this context of frontal operators in the sense of [ 9 ]. Let L be a residuated lattice, and f : L k → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L . We use this characterization of compatible functions in order to prove that (...) the variety of residuated lattices is locally affine complete. We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P ( x , y ) and Q ( x , y ) on a residuated lattice L which imply that the function $${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$$ when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators. (shrink)

A symmetric residuated lattice is an algebra such that is a commutative integral bounded residuated lattice and the equations x=x and =xy are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription εx=x→0. We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive →0)=1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we (...) consider the chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n=1, we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature. (shrink)

This work extend to residuated lattices the results of [7]. It also provides a possible generalization to this context of frontal operators in the sense of [9].Let L be a residuated lattice, and f : Lk → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of residuated lattices (...) is locally affine complete.We study some compatible functions on residuated lattices which are a generalization of frontal operators. We also give conditions for two operations P(x, y) and Q(x, y) on a residuated lattice L which imply that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \mapsto min\{y \in L : P(x, y) \leq Q(x, y)\}}$$\end{document} when defined, is equational and compatible. Finally we discuss the affine completeness of residuated lattices equipped with some additional operators. (shrink)

Let L be a commutative residuated lattice and let f : Lk → L a function. We give a necessary and sufficient condition for f to be compatible with respect to every congruence on L. We use this characterization of compatible functions in order to prove that the variety of commutative residuated lattices is locally affine complete. Then, we find conditions on a not necessarily polynomial function P(x, y) in L that imply that the function x ↦ (...) min{y є L | P(x, y) ⪯ y} is compatible when defined. In particular, Pn(x, y) = yn → x, for natural number n, defines a family, Sn, of compatible functions on some commutative residuated lattices. We show through examples that S1> and S2, defined respectively from P1 and P2, are independent as operations over this variety; i.e. neither S1 is definable as a polynomial in the language of L enriched with S2 nor S2 in that enriched with S1. (shrink)

Logics designed to deal with vague statements typically allow algebraic semantics such that propositions are interpreted by elements of residuated lattices. The structure of these algebras is in general still unknown, and in the cases that a detailed description is available, to understand its significance for logics can be difficult. So the question seems interesting under which circumstances residuated lattices arise from simpler algebras in some natural way. A possible construction is described in this paper.Namely, we (...) consider pairs consisting of a Brouwerian algebra and an equivalence relation. The latter is assumed to be in a certain sense compatible with the partial order, with the formation of differences, and with the formation of suprema of pseudoorthogonal elements; we then call it an s-equivalence relation. We consider operations which, under a suitable additional assumption, naturally arise on the quotient set. The result is that the quotient set bears the structure of a residuated lattice. Further postulates lead to dual BL-algebras. In the case that we begin with Boolean algebras instead, we arrive at dual MV-algebras. (shrink)

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices (...) which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. (shrink)

This is also where we begin investigating lattices of logics and varieties, rather than particular examples.

A quasivariety of algebras has the joint embedding property (JEP) if and only if it is generated by a single algebra A. It is structurally complete if and only if the free ℵ0‐generated algebra in can serve as A. A consequence of this demand, called ‘passive structural completeness’ (PSC), is that the nontrivial members of all satisfy the same existential positive sentences. We prove that if is PSC then it still has the JEP, and if it has the JEP and (...) its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if is relatively semisimple then it is generated by one ‐simple algebra. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC if and only if its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics. (shrink)

This paper discusses Crawley completions of residuated lattices. While MacNeille completions have been studied recently in relation to logic, Crawley completions (i.e. complete ideal completions), which are another kind of regular completions, have not been discussed much in this relation while many important algebraic works on Crawley completions had been done until the end of the 70’s. In this paper, basic algebraic properties of ideal completions and Crawley completions of residuated lattices are studied first in their (...) conncetion with the join infinite distributivity and Heyting implication. Then some results on algebraic completeness and conservativity of Heyting implication in substructural predicate logics are obtained as their consequences. (shrink)

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 (...) defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness. (shrink)

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], H BCK [36] (nowadays called by Ono), etc. In this paper we study the -fragment and the -fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. (...) It results that this notion of fragment is axiomatized by the rules of the sequent calculus for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable. (shrink)

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

A residuated lattice is said to be integrally closed if it satisfies the quasiequations \ and \, or equivalently, the equations \ and \. Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping \\backslash {\mathrm {e}}\) on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed (...) residuated lattices with respect to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic. (shrink)

Bosbach and Riečan states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Riečan states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mureşan through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the (...) present paper, the Glivenko theorem is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Riečan state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei. (shrink)

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the (...) class of integral commutative bounded residuated lattices and to the involutive case. (shrink)

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, (...) residuated modal logic, as well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

Let ${\mathbb{BRL}}$ denote the variety of commutative integral bounded residuated lattices (bounded residuated lattices for short). A Boolean retraction term for a subvariety ${\mathbb{V}}$ of ${\mathbb{BRL}}$ is a unary term t in the language of bounded residuated lattices such that for every ${{\bf A} \in \mathbb{V}, t^{A}}$ , the interpretation of the term on A, defines a retraction from A onto its Boolean skeleton B(A). It is shown that Boolean retraction terms are equationally definable, (...) in the sense that there is a variety ${\mathbb{V}_{t} \subsetneq \mathbb{BRL}}$ such that a variety ${\mathbb{V} \subsetneq \mathbb{BRL}}$ admits the unary term t as a Boolean retraction term if and only if ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ . Moreover, the equation s(x) = t(x) holds in ${\mathbb{V}_{s} \cap \mathbb{V}_{t}}$ . The radical of ${{\bf A} \in \mathbb{BRL}}$ , with the structure of an unbounded residuated lattice with the operations inherited from A expanded with a unary operation corresponding to double negation and a a binary operation defined in terms of the monoid product and the negation, is called the radical algebra of A. To each involutive variety ${\mathbb{V} \subseteq \mathbb{V}_{t}}$ is associated a variety ${\mathbb{V}^{r}}$ formed by the isomorphic copies of the radical algebras of the directly indecomposable algebras in ${\mathbb{V}}$ . Each free algebra in such ${\mathbb{V}}$ is representable as a weak Boolean product of directly indecomposable algebras over the Stone space of the free Boolean algebra with the same number of free generators, and the radical algebra of each directly indecomposable factor is a free algebra in the associated variety ${\mathbb{V}^{r}}$ , also with the same number of free generators.A hierarchy of subvarieties of ${\mathbb{BRL}}$ admitting Boolean retraction terms is exhibited. (shrink)

This work presents a model-theoretic approach to the study of the amalgamation property for varieties of semilinear commutative residuated lattices. It is well-known that if a first-order theory T enjoys quantifier elimination in some language L, the class of models of the set of its universal consequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm T_\forall}$$\end{document} has the amalgamation property. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} be the theory of an (...) elementary subclass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}}$$\end{document} of the linearly ordered members of a variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} of semilinear commutative residuated lattices. We show that whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} has elimination of quantifiers, and every linearly ordered structure in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} is a model of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}_\forall(\mathbb{K})}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} has the amalgamation property. We exploit this fact to provide a purely model-theoretic proof of amalgamation for particular varieties of semilinear commutative residuated lattices. (shrink)

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬ x ∨ ¬¬ x = 1. Some applications are given.

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of (...) these operators are presented. (shrink)

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17—19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five (...) 3-potent commutative representable atoms and two integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members. (shrink)

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of (...) residuated lattices leads us to study other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c. If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C M is isomorphic to C e M, and C e is adjoint to, where assigns to an object A of IRL0 the product A × A 0 which is an object of NRL. (shrink)

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 + is also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product. (shrink)

For arbitrary finite group $G$ and countable Dedekind domain $R$ such that the residue field $R/P$ is finite for every maximal $R$ -ideal $P$ , we show that the localizations at every maximal ideal of two $RG$ -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary $R$ -torsion-free $RG$ -modules and we apply the previous results to show the decidability of the (...) theory of ${\vec Z}C(2)^2$ -lattices. Eventually, we show that ${\vec Z} [i] C(2)^2$ -lattices have undecidable theory. (shrink)

Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of (...) which is a copy of the {, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops). (shrink)

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬x ∨ ¬¬x = 1. Some applications are given.

Much work has been done on specific instances of residuated lattices with modal operators . In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian ℓ-groups with conuclei, and negative cones of ℓ-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes (...) the role played by reducts in the proofs of these categorical equivalences. Lastly, we develop a connection between translations of logics and images of modal operators. (shrink)

The variety of residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., \-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated \-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated \-groupoids, their ideals, and develop a theory of (...) left nuclei. Finally, we extend some parts of the theory of join-completions of residuated \-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices. (shrink)

We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. (...) We compare our results with other works and show some applications of the equivalence. (shrink)

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural (...) logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics. Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved. (shrink)

We study the sequent system mentioned in the author's work as CyInFL with ‘intuitionistic’ sequents. We explore the connection between this system and symmetric constructive logic of Zaslavsky 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 (...) FLew‐algebras. In particular, we prove the definitional equivalence of symmetric constructive FLewc‐algebras (algebraic models of symmetric constructive logic) and Nelson FLew‐algebras (algebras introduced by Spinks and Veroff, 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. As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL‐algebras are decidable. (shrink)

At present, person reidentification based on attention mechanism has attracted many scholars’ interests. Although attention module can improve the representation ability and reidentification accuracy of Re-ID model to a certain extent, it depends on the coupling of attention module and original network. In this paper, a person reidentification model that combines multiple attentions and multiscale residuals is proposed. The model introduces combined attention fusion module and multiscale residual fusion module in the backbone network ResNet 50 to enhance the feature flow (...) between residual blocks and better fuse multiscale features. Furthermore, a global branch and a local branch are designed and applied to enhance the channel aggregation and position perception ability of the network by utilizing the dual ensemble attention module, as along as the fine-grained feature expression is obtained by using multiproportion block and reorganization. Thus, the global and local features are enhanced. The experimental results on Market-1501 dataset and DukeMTMC-reID dataset show that the indexes of the presented model, especially Rank-1 accuracy, reach 96.20% and 89.59%, respectively, which can be considered as a progress in Re-ID. (shrink)

In this work we study the decidability of a class of global modal logics arising from Kripke frames evaluated over certain residuated lattices, known in the literature as modal many-valued logics. We exhibit a large family of these modal logics which are undecidable, in contrast with classical modal logic and propositional logics defined over the same classes of algebras. This family includes the global modal logics arising from Kripke frames evaluated over the standard Łukasiewicz and Product algebras. We (...) later refine the previous result, and prove that global modal Łukasiewicz and Product logics are not even recursively axiomatizable. We conclude by closing negatively the open question of whether each global modal logic coincides with its local modal logic closed under the unrestricted necessitation rule. (shrink)

Local item dependence is one of the most critical assumption in the Rasch model when it comes to the validity of a test. As the field of vocabulary assessment is calling for more clarity and validity for vocabulary tests, such assumption becomes more important than ever. The article offers a Rasch-based investigation into the issue of LID with the focus on the two popular formats of Vocabulary Levels Tests : multiple-choice and matching. A Listening Vocabulary Levels Test and an (...) Updated Vocabulary Levels Test were given to a single cohort of 311 university students in an English as a Foreign Language context. The analyses of raw score and standardized residuals correlations were conducted. The findings found no relationship between the 4-option, multiple-choice format of the LVLT and item local dependence. However, results from score and standardized residuals correlations analyses showed a strong link between the 3-item-per-cluster, matching format and item local dependence. The study calls for greater attention to the format of future vocabulary tests and support the use of meaning-recall formats in vocabulary testing. (shrink)

In the paper we introduce two conditions and ) which are strengthenings of Birkhoff’s conditions. We prove that an upper continuous and strongly atomic lattice is distributive if and only if it satisfies and ). This result extends a theorem of R.P. Dilworth characterizing distributivity in terms of local distributivity and a theorem of M. Ward characterizing distributivity by means of covering diamonds.

ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests (...) that a reduction of the first to the latter may be possible, encoding Positive Lattice Logic as a fragment of Two-Sorted, Residuated Modal Logic. The reduction is analogous to the well-known Gödel-McKinsey-Tarski translation of Intuitionistic Logic into the S4 system of normal modal logic. In this article, we carry out this reduction in detail and we derive some properties of PLL from corresponding properties of First-Order Logic. The reduction we present is extendible to the case of lattices with operators, making use of recent results by this author on the relational representation of normal lattice expansions. (shrink)

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.

Let $\mathcal{M}$ be an o-minimal expansion of a real closed field $R$. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of $\langle R^{n},+\rangle$ contains a definable generic set and therefore admits a lattice.

The category \(\mathbb {DRDL}{'}\), whose objects are c-differential residuated distributive lattices satisfying the condition \(\textbf{CK}\), is the image of the category \(\mathbb {RDL}\), whose objects are residuated distributive lattices, under the categorical equivalence \(\textbf{K}\) that is constructed in Castiglioni et al. (Stud Log 90:93–124, 2008). In this paper, we introduce weak monadic residuated lattices and study some of their subvarieties. In particular, we use the functor \(\textbf{K}\) to relate the category \(\mathbb {WMRDL}\), whose objects (...) are weak monadic residuated distributive lattices, and the category \(\mathbb {WMDRDL}{'}\), whose objects are pairs formed by an object of \(\mathbb {DRDL}{'}\) and a center weak universal quantifier. (shrink)