Commutative Residuated Lattices Research Articles

Archimedean classes in integral commutative residuated chains

AbstractThis paper investigates a quasi‐variety of representable integral commutative residuated lattices axiomatized by the quasi‐identity resulting from the well‐known Wajsberg identity (p → q) → q ≤ (q → p) → p if it is written as a quasi‐identity, i. e., (p → q) → q ≈ 1 ⇒ (q → p) → p ≈ 1. We prove that this quasi‐identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi‐variety is in fact a variety and provide an axiomatization. The obtained results shed some light on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL‐algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL‐algebras is generated by its Archimedean members (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Residuated Lattices as an Algebraic Semantics for Paraconsistent Nelson's Logic

The class of NPc-lattices is introduced as a quasivariety of commutative residuated lattices, and it is shown that the class of pairs (A,A+) such that A is an NPc-lattice and A+ is its positive cone, is a matrix semantics for Nelson paraconsistent logic.

FREE REPRESENTABLE IDEMPOTENT COMMUTATIVE RESIDUATED LATTICES

The structure of simple and directly indecomposable members of the variety RICRL of representable idempotent commutative residuated lattices is described. These descriptions are employed to determine the size and structure of free algebras in RICRL, using existing theory on the Fraser–Horn and Apple properties. A formula for the free spectrum of RICRL is given.

Partial residuated structures and quantum structures

Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics--classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.

Compatible operations on commutative residuated lattices

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.

Structure of commutative cancellative integral residuated lattices on (0, 1

ΠMTL-algebras were introduced as an algebraic counterpart of the cancellative extension of monoidal t-norm based logic. It was shown that they form a variety generated by ΠMTL-chains on the real interval [0, 1]. In this paper the structure of these generators is investigated. The results illuminate the structure of cancellative integral commutative residuated chains, because every such algebra belongs to the quasivariety generated by the zero-free subreducts on (0, 1] of all ΠMTL-chains on [0, 1].

Substructural fuzzy logics

AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

Implicational subreducts of n-potent commutative residuated lattices

The implicational subreducts of n-potent commutative integral residuated lattices are axiomatized using a new embedding of a BCK-algebra into a commutative integral residuated lattice. The class of {→, 1, ≤ }-subreducts of commutative residuated lattices satisfying xn ≤ xm is also axiomatized, as are other subreduct classes.

Representable idempotent commutative residuated lattices

It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of positive relevance logic containing the mingle and Godel-Dummett axioms has a solvable deducibility problem.

Equational Bases for Joins of Residuated-lattice Varieties

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.

THE STRUCTURE OF COMMUTATIVE RESIDUATED LATTICES

A commutative residuated lattice, is an ordered algebraic structure [Formula: see text], where (L, ·, e) is a commutative monoid, (L, ∧, ∨) is a lattice, and the operation → satisfies the equivalences [Formula: see text] for a, b, c ∊ L. The class of all commutative residuated lattices, denoted by [Formula: see text], is a finitely based variety of algebras. Historically speaking, our study draws primary inspiration from the work of M. Ward and R. P. Dilworth appearing in a series of important papers [9, 10, 19–22]. In the ensuing decades special examples of commutative, residuated lattices have received considerable attention, but we believe that this is the first time that a comprehensive theory on the structure of residuated lattices has been presented from the viewpoint of universal algebra. In particular, we show that [Formula: see text] is an "ideal variety" in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, we present an equational basis for the subvariety [Formula: see text] generated by all commutative, residuated chains. We conclude the paper by proving that the congruence lattice of each member of [Formula: see text] is an algebraic, distributive lattice whose meet-prime elements form a root-system (dual tree). This result, together with the main results in [12, 18], will be used in a future publication to analyze the structure of finite members of [Formula: see text]. A comprehensive study of, not necessarily commutative, residuated lattices is presented in [4].

A proof of standard completeness for Esteva and Godo's logic MTL

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo's logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.