The subvariety lattice for representable idempotent commutative residuated lattices

RICRL denotes the variety of commutative residuated lattices which have an idempotent monoid operation and are representable in the sense that they are subdirect products of linearly ordered algebras. It is shown that the subvariety lattice of RICRL is countable, despite its complexity and in contrast to several varieties of closely related algebras.     

Cancellative residuated lattices arising on 2-generated submonoids of natural numbers

It is known that there are only two cancellative atoms in the subvariety lattice of residuated lattices, namely the variety of Abelian ?-groups ${\mathcal{CLG}}$ generated by the additive ?-group of integers and the variety ${\mathcal{CLG}^-}$ generated by the negative cone of this ?-group. In this paper we consider all cancellative residuated chains arising on 2-generated submonoids of natural numbers and show that almost all of them generate a cover of ${\mathcal{CLG}^-}$ . This proves that there are infinitely many covers above ${\mathcal{CLG}^-}$ which are commutative, integral, and representable.     

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 ${\mathbb{K}}$ of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of ${\mathbb{K}}$ , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in ${\mathbb{K}}$ , and we analyze the subvariety of representable algebras in ${\mathbb{K}}$ . Finally, we consider some specific class of bounded integral commutative residuated lattices ${\mathbb{G}}$ , and for each fixed element ${{\bf L} \in \mathbb{G}}$ , 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.     

Residuated Structures,Concentric Sums and Finiteness Conditions

This article is motivated by a concern with finiteness conditions on varieties of residuated structures—particularly residuated meet semilattice-ordered commutative monoids. A “concentric sum” construction is developed and is used to prove, among other results, a local finiteness theorem for a class that encompasses all n-potent hoops and all idempotent subdirect products of residuated chains. This in turn implies that a range of residuated lattice-based varieties have the finite embeddability property, whence their quasi-equational theories are decidable. Applications to substructural logics are discussed.     

Craig interpolation for semilinear substructural logics

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.     

全序幂等元剩余格

研究了全序幂等元剩余格,给出了全序幂等元剩余格的另一种新的构造方法,并且得到了这类剩余格的结构定理,推广了相关文献的结论．     

Generalized Bosbach states: Part II

We continue the investigation of generalized Bosbach states that we began in Part I, restricting our research to the commutative case and treating further aspects related to these states. Part II is concerned with similarity convergences, continuity of states and the construction of the s-completion of a commutative residuated lattice, where s is a generalized Bosbach state.     

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies. We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved. Received August 21, 2005; accepted in final form October 17, 2006.     

Commutative idempotent residuated lattices

We investigate the variety of residuated lattices with a commutative and idempotent monoid reduct.     

The structure of idempotent residuated chains

In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains. This work is supported by a grant of NSF, China # 10471112 and a grant of Shaanxi Provincial Natural Science Foundation # 2005A15.     

Bounded BCK‐algebras and their generated variety

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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aglianò-Montagna type decomposition of linear pseudo hoops and its applications

We decompose every linear pseudo hoop as an Aglianò-Montagna type of ordinal sum of linear Wajsberg pseudo hoops which are either negative cones of linear ?-groups or intervals in linear unital ?-groups with strong unit. We apply the decomposition to present a new proof that every linear pseudo BL-algebra and consequently every representable pseudo BL-algebra is good. Moreover, we show that every maximal filter and every value of a linear pseudo hoop is normal, and every σ-complete linear pseudo hoop is commutative.     

相似剩余格及其对应逻辑系统的完备性

引入了相似剩余格的概念,讨论了剩余格上相似算子和等价算子的关系,并得到了真值剩余格和相似剩余格相互转化的方法.其次,研究了相似剩余格上的相似滤子,利用相似滤子刻画了可表示的相似剩余格.最后,引入了相似剩余格对应的逻辑系统,证明了其完备性定理,并得到了其成为半线性逻辑的条件.     

On the structure of generalized BL-algebras

A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities . It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras. This paper is dedicated to Walter Taylor. Received March 7, 2005; accepted in final form July 25, 2005.     

Conical residuated lattice-ordered idempotent monoids

In this paper, we study some special residuated lattices, namely, conical idempotent residuated lattices. After obtaining some properties of such residuated lattices, we establish a structure theorem for conical idempotent residuated lattices. This work is supported by a grant of NSF, China # 10471112 and a grant of Shaanxi Provincial Natural Science Foundation # 2005A15.     

Varieties of idempotent semirings with commutative addition

The multiplicative reduct of an idempotent semiring with commutative addition is a regular band. Accordingly there are 13 distinct varieties consisting of idempotent semirings with commutative addition corresponding to the 13 subvarieties of the variety of regular bands. The lattice generated by the these 13 semiring varieties is described and models for the semirings free in these varieties are given. Received April 22, 2004; accepted in final form June 3, 2005.     

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.     

Weak effect algebras

Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is restricted to a partial cancellative operation. We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known structure theory of BL-algebras. Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups. This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).     

On the finite embeddability property for residuated ordered groupoids

The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman's finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the FEP as well. The same holds for their respective subclasses of (bounded) (semi-)lattice ordered structures. The assumption of integrality cannot be dropped in general--the class of commutative, residuated, lattice ordered monoids does not have the FEP--but the class of -potent commutative residuated lattice ordered monoids does have the FEP, for any .