Integral Residuated Lattices Research Articles

Structural and universal completeness in algebra and logic

In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.

Projectivity in (bounded) commutative integral residuated lattices

Projectivity in (bounded) commutative integral residuated lattices

From partially ordered monoids to partially ordered groups via free nuclear preimages

Two fundamental constructions operating on residuated lattices and partially ordered monoids (pomonoids) are nuclear images and conuclear images. Nuclear images allow us to construct many of the ordered algebras which arise in non-classical logic (such as pomonoids, semilattice-ordered monoids, and residuated lattices) from cancellative ordered algebras. Conuclear images then allow us to construct some of these cancellative algebras from partially ordered or lattice-ordered groups. Among other things, we show that finite (commutative) integral residuated lattices are precisely the finite nuclear images of (commutative) cancellative integral residuated lattices and that (commutative) integrally closed pomonoids are precisely the nuclear images of subpomonoids of partially ordered (Abelian) groups. The key construction is the free nuclear preimage of a pomonoid. As a by-product of our study of free nuclear preimages, we obtain a syntactic characterization of quasivarieties of pomonoids and semilattice-ordered monoids closed under nuclear images.

A lattice-theoretical approach to extensions of filters in algebras of substructural logic

Commutative bounded integral residuated lattices (residutaed lattices, in short) form a large class of algebras containing algebras which are algebraic counterparts of certain propositional fuzzy logics. The paper deals with the so-called extended filters of filters of residuated lattices. It is used the fact that the extended filters of filters associated with subsets coincide with those associated ones with corresponding filters. This makes it possible to investigate the set of all extended filters of residuated lattices within the Heyting algebras of their filters by means of the structural methods of the theory of such algebras.

Fragments of Quasi-Nelson: The Algebraizable Core

Abstract This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.

Splittings in varieties of logic

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

Quasi-Nelson algebras and fragments

AbstractThe variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Adjoint Operations in Twist-Products of Lattices

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

Splittings in GBL-algebras I: The general case

We study and characterize splitting algebras in varieties of integral residuated (semi)lattices; the main result is a complete characterization of the splitting algebras in the variety of GBLew-algebras, i.e. integral, bounded, commutative and divisible residuated lattices.

A Lindström theorem in many-valued modal logic over a finite MTL-chain

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

Kites and residuated lattices

We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.

Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G operations as well as expansions of some commutative integral residuated lattices with successor operations.

Notes on “Some Properties of <i>L</i>-fuzzy Approximation Spaces on Bounded Integral Residuated Lattices”

In this note, we continue the works in the paper [Some properties of <i>L</i>-fuzzy approximation spaces on bounded integral residuated lattices", Information Sciences, 278, 110-126, 2014]. For a complete involutive residuated lattice, we show that the <i>L</i>-fuzzy topologies generated by a reflexive and transitive <i>L</i>-relation satisfy (TC) _<i>L</i> or (TC) _<i>R</i> axioms and the <i>L</i>-relations induced by two <i>L</i>-fuzzy topologies, which are generated by a reflexive and transitive <i>L</i>-relation, are all the original <i>L</i>-relation; and give out some conditions such that the <i>L</i>-fuzzy topologies generated by two <i>L</i>-relations, which are induced by an <i>L</i>-fuzzy topology, are all the original <i>L</i>-fuzzy topology.

Regular elements and Kolmogorov translation in residuated lattices

In this article, we study in detail the regular elements of a bounded, commutative and integral residuated lattice. We introduce the notion of a regular variety and explore its relationship with the Kolmogorov negative translation. In addition, we investigate the corresponding notions in the axiomatic extensions of the Full Lambek Calculus with exchange and weakening.

Lattice-theoretic properties of algebras of logic

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.

Some properties of L-fuzzy approximation spaces based on bounded integral residuated lattices

In this paper, we present a general framework for the study of L-fuzzy rough sets based on bounded integral residuated lattices. We introduce and discuss the notions of L-fuzzy topological spaces, left (right) lower and left (right) upper L-fuzzy approximation operators based on a complete bounded integral residuated lattice, investigate union, intersection, and composition operations of L-fuzzy approximation spaces, and study some properties of the L-fuzzy topologies generated by these L-fuzzy rough approximation operators and the L-relations induced by an L-fuzzy topology.

The subvariety of commutative residuated lattices represented by twist-products

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety {Mathematical expression} of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of {Mathematical expression}, a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in {Mathematical expression}, and we analyze the subvariety of representable algebras in {Mathematical expression}. Finally, we consider some specific class of bounded integral commutative residuated lattices {Mathematical expression}, and for each fixed element {Mathematical expression}, we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras. © 2014 Springer Basel.

Cayley’s and Holland’s Theorems for Idempotent Semirings and Their Applications to Residuated Lattices

We extend Cayley’s and Holland’s representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integral residuated lattices and many of its subvarieties.

On a Definition of a Variety of Monadic ℓ-Groups

In this paper we expand previous results obtained in [2] about the study of categorical equivalence between the category IRL 0 of integral residuated lattices with bottom, which generalize MV-algebras and a category whose objects are called c-differential residuated lattices. The equivalence is given by a functor $${{\mathsf{K}^\bullet}}$$ , motivated by an old construction due to J. Kalman, which was studied by Cignoli in [3] in the context of Heyting and Nelson algebras. These results are then specialized to the case of MV-algebras and the corresponding category $${MV^{\bullet}}$$ of monadic MV-algebras induced by "Kalman's functor" $${\mathsf{K}^\bullet}$$ . Moreover, we extend the construction to l-groups introducing the new category of monadic l-groups together with a functor $${\Gamma ^\sharp}$$ , that is "parallel" to the well known functor $${\Gamma}$$ between l and MV-algebras.

&lt;i&gt;L&lt;/i&gt;-Topological Spaces Based on Residuated Lattices

In this paper, we introduce the notion of L-topological spaces based on a complete bounded integral residuated lattice and discuss some properties of interior and left (right) closure operators.