On v-filters and normal v-filters of a residuated lattice with a weak vt-operator

In this paper, we study further the filter theory of residuated lattices. First, we discuss the concepts of filters and normal filters of residuated lattices and propose some equivalent conditions for them. Then we introduce and investigate the notions of v-filter and normal v-filter of a residuated lattice with a weak vt-operator and lay bare the formulas for calculating the v-filters and the normal v-filters generated by subsets. Finally we show that the lattices of v-filters and normal v-filters of a residuated lattice with a vt-operator are both complete Brouwerian lattices.     

Relationships between generalized Bosbach states and <Emphasis Type="Italic">L</Emphasis>-filters on residuated lattices

Generalized Bosbach states and filters on residuated lattices have been extensively studied in the literature. In this paper, relationships between generalized Bosbach states and residuated-lattice-valued filters, also called L-filters, on residuated lattices are investigated. Particularly, type I and type II L-filters and their subclasses are defined, and some their properties are obtained. Then relationships between special types of L-filters and the generalized Bosbach states are considered where generalized Bosbach states are characterized by some type I or type II L-filters with additional conditions. Associated with these relationships, new subclasses of generalized Bosbach states such as implicative type IV, V, VI states, fantastic type IV states and Boolean type IV states are introduced, and the relationships between various types of generalized Bosbach states are investigated in detail. In particular, the existence of several generalized Bosbach states is provided and, as application, some typical subclasses of residuated lattices such as Rl-monoids, Heyting algebras and Boolean algebras are characterized by these generalized Bosbach states.     

(?, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring

In this paper, we consider a ring as a universal set and study (?, T)-fuzzy rough approximation operators with respect to a TL-fuzzy ideal of a ring. First, some new properties of generalized (?, T)-fuzzy rough approximation operators are obtained. Then, a new fuzzy algebraic structure - TL-fuzzy rough ideal is defined and its properties investigated. And finally, the homomorphism of (?, T)-fuzzy rough approximation operators is studied.     

A Generalization of Local Fuzzy Structures

The class of bounded residuated lattice ordered monoids Rl-monoids) contains as proper subclasses the class of pseudo BL-algebras (and consequently those of pseudo MV-algebras, BL-algebras and MV-algebras) and of Heyting algebras. In the paper we introduce and investigate local bounded Rl-monoids which generalize local algebras from the above mentioned classes of fuzzy structures. Moreover, we study and characterize perfect bounded Rl-monoids.     

Lattice implication ordered semigroups

From the viewpoint of semantics, lattice implication algebras provide a basis to establish lattice-valued logic with truth value in a relatively general lattice. In this paper, we first introduce two notions of lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, which induced by lattice implication algebras. Secondly, we study some of their basic properties and prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice ordered semigroup. We also define the homomorphism mapping between lattice implication n-ordered semigroups. Finally, we discuss some properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups.     

Algebraic properties of LA-languages

In this study, we introduce the concepts of L-valued regular substitution (LA-substitution), deterministic L-valued regular substitution (DLA-substitution), L-valued fuzzy homomorphism and its inverse images, homomorphism and its inverse images for a lattice-ordered monoid L. We also study the properties of LA-languages and DLA-languages under the above-mentioned algebraic operations. The algebraic characterization of the L-valued regular language is given.     

BR0代数的关联MP滤子和布尔MP滤子

引入BR0代数的关联胛滤子。研究它的特征,证明如下主要结果：F是关联脚滤子,当且仅当F是布尔仰滤子。从而BR0代数成为Boole代数的充要条件是每个胛滤子均为布尔胛滤子或关联肿滤子。     

The pseudo-linear semantics of interval-valued fuzzy logics

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct triangle logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs.In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices.As a consequence, we are able to prove the pseudo-chain completeness of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-norm based logic (MTL).This result also provides a better insight in the structure of triangle algebras; it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L,α), in which L is a residuated lattice and α an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on L and α.     

Algebraic properties of complete residuated lattice valued tree automata

This paper investigates tree automata based on complete residuated lattice valued (referred to as L-valued) logic. First, we define the notions of L-valued set of pure subsystems and L-valued set of strong pure subsystems, as well as, their relation is considered. Also, L-valued n-tuple operator consist of n successors is defined, some of its properties are examined and its relation with pure subsystem is analyzed. Furthermore, we investigate some concepts such as L-valued set of (strong) homomorphisms, L-valued set of (strong) isomorphisms, and L-valued set of admissible relations. Moreover, we discuss bifuzzy topological characterization of L-valued tree automata. Finally, the relations of homomorphisms between the L-valued tree automata to continuous mappings and open mappings is examined.     

L-fuzzy numbers and their properties

In this paper, the notions of L-fuzzy convex sets and L-fuzzy numbers are introduced where L is a completely distributive lattice. The notions of [0, 1]-fuzzy convex sets and [0, 1]-fuzzy numbers are generalized. Furthermore their properties and characterizations are presented in terms of cut sets of L-fuzzy sets.     

Exponential p-stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays

In this paper, we study the impulsive stochastic Cohen–Grossberg neural networks with mixed delays. By establishing an L-operator differential inequality with mixed delays and using the properties of M-cone and stochastic analysis technique, we obtain some sufficient conditions ensuring the exponential p-stability of the impulsive stochastic Cohen–Grossberg neural networks with mixed delays. These results generalize a few previous known results and remove some restrictions on the neural networks. Two examples are also discussed to illustrate the efficiency of the obtained results.     

Lattice structure of e-implications on L

In this paper, we consider the set of all e-implications on L^∗ and define two operations on the set of all representable e-implications on L^∗, thus endowing it with monoid structure which is also a distributive complete lattice.     

Intuitionistic fuzzy Hv-submodules

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov is one among them. In this paper, we apply the concept of an intuitionistic fuzzy set to H_v-modules. The notion of an intuitionistic fuzzy H_v-submodule of an H_v-module is introduced, and some related properties are investigated. Characterizations of intuitionistic fuzzy H_v-submodules are given.     

Face recognition by sparse discriminant analysis via joint L2,1-norm minimization

Recently, joint feature selection and subspace learning, which can perform feature selection and subspace learning simultaneously, is proposed and has encouraging ability on face recognition. In the literature, a framework of utilizing L_2,1-norm penalty term has also been presented, but some important algorithms cannot be covered, such as Fisher Linear Discriminant Analysis and Sparse Discriminant Analysis. Therefore, in this paper, we add L_2,1-norm penalty term on FLDA and propose a feasible solution by transforming its nonlinear model into linear regression type. In addition, we modify the optimization model of SDA by replacing elastic net with L_2,1-norm penalty term and present its optimization method. Experiments on three standard face databases illustrate FLDA and SDA via L_2,1-norm penalty term can significantly improve their recognition performance, and obtain inspiring results with low computation cost and for low-dimension feature.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

The stable topology for residuated lattices

The spectrum of a residuated lattice L is the set Spec(L) of all prime i-filters. It is well known that Spec(L) can be endowed with the spectral topology. The main scope of this paper is to introduce and study another topology on Spec(L),?the so called stable topology, which turns out to be coarser than the spectral one. With this and in view, we introduce the notions of pure i-filter for a residuated lattice and the notion of normal residuated lattice. So, we generalize to case of residuated lattice some results relative to MV-algebras (Belluce and Sessa in Quaest Math 23:269–277, 2000; Cavaccini et?al. in Math Japonica 45(2):303–310, 1997) or BL-algebras (Eslami and Haghani in Kybernetika 45:491–506, 2009; Leustean in Central Eur J Math 1(3): 382–397, 2003; Turunen and Sessa in Mult-Valued Log 6(1–2):229–249, 2001).     

The radical of a perfect residuated structure

In this paper we extend some properties of the radical of an MTL-algebra to the non-commutative case of a more general residuated structure, namely the FL_w-algebra. For the particular case of pseudo-MTL algebras, some specific results are presented. We introduce the notion of a local additive measure on a perfect pseudo-MTL algebra and we prove that, with some additional conditions, every local additive measure can be extended to a Rie?an state; a necessary and sufficient condition is given for such an extension to be a Bosbach state.     

States on semi-divisible residuated lattices

Given a residuated lattice L, we prove that the subset MV(L) of complement elements x ^* of L generates an MV-algebra if, and only if L is semi-divisible. Riečan states on a semi-divisible residuated lattice L, and Riečan states on MV(L) are essentially the very same thing. The same holds for Bosbach states as far as L is divisible. There are semi-divisible residuated lattices that do not have Bosbach states. These results were obtained when the authors visited Academy of Science, Czech Republic, Institute of Comp. Sciences in Autumn 2006.     

l2-l∞ filter design for discrete-time singular Markovian jump systems with time-varying delays

The problem of l₂-l_∞ filter design for discrete-time singular systems with Markovian jump and time-varying delays is investigated in this paper. By using the delay partitioning technique, a delay-dependent condition is established to guarantee the filtering error systems to be stochastically admissible and achieve a prescribed l₂-l_∞ performance index. Based on the derived condition, the full-order and reduced-order filters with mode-independent characterization are designed in a unified framework. The corresponding filter parameters can be obtained by solving a set of linear matrix inequalities. The reported results not only depend upon the delay, but also depend upon the partitioning, which aims at reducing the conservatism. Two numerical examples are provided to illustrate the effectiveness of the proposed methods.     

On probabilistic n-ary hypergroups

The T-fuzzy n-ary subhypergroups of an n-ary hypergroup are defined by using triangular norms and some related properties are hence obtained. In particular, we consider the probabilistic version of n-ary hypergroups by using random sets and show that the fuzzy n-ary hypergroups defined by triangular norms are consequences of some probabilistic n-ary hypergroups under certain conditions. Some results on n-ary hypergroups recently given by Davvaz and Corsini are extended.