Interior and closure operators on bounded residuated lattice ordered monoids

GMV-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior GMV-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on DRl-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on GMV-algebras.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

Modal operators on commutative residuated lattices

We prove some fundamental properties of monotone modal operators on bounded commutative integral residuated lattices (CRL). Moreover we give a positive answer to the problem left open in [RACHŮNEK, J.—ŠALOUNOV á, D.: Modal operators on bounded commutative residuated Rℓ-monoids, Math. Slovaca 57 (2007), 321–332].     

On minimal residuated bounded mappings in atomistic lattices

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

Valuations and closure operators on finite lattices

Let L be a lattice. A function f:L→R (usually called evaluation) is submodular if f(x∧y)+f(x∨y)≤f(x)+f(y), supermodular if f(x∧y)+f(x∨y)≥f(x)+f(y), and modular if it is both submodular and supermodular. Modular functions on a finite lattice form a finite dimensional vector space. For finite distributive lattices, we compute this (modular) dimension. This turns out to be another characterization of distributivity (Theorem 3.9). We also present a correspondence between isotone submodular evaluations and closure operators on finite lattices (Theorem 5.5). This interplay between closure operators and evaluations should be understood as building a bridge between qualitative and quantitative data analysis.     

Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of to the lattice of subvarieties of . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.     

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.     

Commutative idempotent residuated lattices

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

Constructing simple residuated lattices

A method of constructing residuated lattices is presented. As an application, examples of simple, integral, cancellative, distributive residuated lattices are given that are not linearly ordered. This settles a problem raised in [5] and [2].     

Minimal varieties of residuated lattices

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated bounded-lattice.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 1, 2003; accepted in final form April 27, 2004.     

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．     

Relation equations in residuated lattices

In questa nota si affronta il problema della risoluzione di equazioni matriciali del tipoAX=B, doveA eB sono matrici a valori su un reticolo distributivo residuato rispetto a una moltiplicazione. In particolare, si individua la più grande soluzione di una tale equazione e si danno condizioni relative alle soluzioni minimali.     

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.

     

Central points and approximation in residuated lattices

The paper presents results on approximation in residuated lattices given that closeness is assessed by means of biresiduum. We describe central points and optimal central points of subsets of residuated lattices and examine several of their properties. In addition, we present algorithms for two problems regarding optimal approximation.     

Fixed elements in involutive residuated lattices

An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element d. The involution, i.e., the function \({x \mapsto x\backslash d}\), of an IRL induces a lattice anti-isomorphism, and is also an order-2 bijection of the underlying set. We examine which such bijections may be induced by the involution of an IRL.     

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.     

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.