Commutative idempotent residuated lattices

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

On minimal residuated bounded mappings in atomistic lattices

Relationship between automorphisms and residuated bounded mappings in atomistic lattices is studied.      

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.     

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.     

Archimedean classes in integral commutative residuated chains

This 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)     

Representable idempotent commutative residuated lattices

It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The -generated subdirectly irreducible algebras in this variety are shown to have at most 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 Gödel-Dummett axioms has a solvable deducibility problem.

     

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 .

     

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.     

剩余格的模糊滤子理论

运用模糊集的方法和原理进一步深入研究剩余格的滤子问题.在剩余格中引入了模糊预线性滤子,模糊可除滤子和模糊Glivenko滤子三类新的模糊滤子概念,给出了它们的若干性质和等价刻画.系统讨论了这三类模糊滤子以及模糊正关联滤子,模糊Boolean滤子,模糊MV滤子和模糊正则滤子间的相互关系,证明了一个模糊滤子为模糊MV滤子当且仅当它既是模糊正则滤子又是模糊可除滤子的结论.     

关于蕴涵BCK—代数的伴随半群

我们证明了蕴涵BCK-代数的伴随半群是一个上半格;具有条件（s)的蕴涵BCK-代数的伴随半群是一个广义布尔代数。更进一步证明了有界蕴涵BCK-代数的伴随半群是一个布尔代数。     

Subdirectly irreducible sectionally pseudocomplemented semilattices

Sectionally pseudocomplemented semilattices are an extension of relatively pseudocomplemented semilattices—they are meet-semilattices with a greatest element such that every section, i.e., every principal filter, is a pseudocomplemented semilattice. In the paper, we give a simple equational characterization of sectionally pseudocomplemented semilattices and then investigate mainly their congruence kernels which leads to a characterization of subdirectly irreducible sectionally pseudocomplemented semilattices. Supported by the Council of the Czech Government, MSM 6198959214.     

Induced residuated maps

The purpose of this paper is to introduce a class of mappings from a lattice L, whose elements are residuated maps, into itself. The main results of this paper identify certain injective residuated mappings of L and order automorphisms of a sublattice of L with mappings from this class.     

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．     

Local bounded commutative residuated ℓ-monoids

Bounded commutative residuated lattice ordered monoids (Rℓ-monoids) are a common generalization of, e.g., BL-algebras and Heyting algebras. In the paper, the properties of local and perfect bounded commutative Rℓ-monoids are investigated.     

Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Grätzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.     

Notes on sectionally complemented lattices. III

Summary In a recent survey article, G. Grätzer and E. T. Schmidt raise the problem when is the ideal lattice of a sectionally complemented chopped lattice sectionally complemented. The only general result is a 1999 lemma of theirs, stating that if the finite chopped lattice is the union of two ideals that intersect in a two-element ideal U, then the ideal lattice of M is sectionally complemented. In this paper, we present examples showing that in many ways their result is optimal. A typical result is the following: For any finite sectionally complemented lattice U with more than two elements, there exists a finite sectionally complemented chopped lattice M that is (i) the union of two ideals intersecting in the ideal U; (ii) the ideal lattice of M is not sectionally complemented.     

On Matroids and Orlik-Solomon Algebras

In this paper we axiomatize combinatorics of arrangements of affine hyperplanes, which is a generalization of matroids, called quasi-matroids. We show that quasi-matroids are equivalent to pointed matroids. On the other hand, the Orlik-Solomon (OS) algebra of a quasimatroid can be constructed. We prove that the OS algebra of a quasi-matroid is isomorphic to the direct image of the OS algebra of a matroid by the linear derivation.AMS Subject Classification: 03B35, 13D03, 52C35.     

On symmetric extended MS-algebras whose congruences are permutable

Algebras whose congruences are permutable were investigated by a number of authors in the literature. In this paper, we study the symmetric extended MS-algebras whose congruences are permutable. Some results obtained by Jie Fang on symmetric extended De Morgan algebras are generalized.