Hybrid generalized Bosbach and Rie c̆ an states on non-commutative residuated lattices

Generalized Bosbach and Rie c? an states, which are useful for the development of an algebraic theory of probabilistic models for commutative or non-commutative fuzzy logics, have been investigated in the literature. In this paper, a new way arising from generalizing residuated lattice-based filters from commutative case to non-commutative one is applied to introduce new notions of generalized Bosbach and Rie c? an states, which are called hybrid ones, on non-commutative residuated lattices is provided, and the relationships between hybrid generalized states and those existing ones are studied, examples show that they are different. In particular, two problems from L.C. Ciungu, G. Georgescu, and C. Mure, “Generalized Bosbach States: Part I” (Archive for Mathematical Logic 52 (2013):335–376) are solved, and properties of hybrid generalized states, which are similar to those on commutative residuated lattices, are obtained without the condition “strong”.     

States on finite monoidal t-norm based algebras

The aim of this paper is to study the existences of Bosbach states and Rie?an states on finite monoidal t-norm based algebras (MTL-algebras for short). We give some examples to show that there exist MTL-algebras having no Bosbach states and Rie?an states. The conditions under which MTL-algebras have Bosbach states and Rie?an states are investigated, respectively. We prove that Rie?an states on MTL-algebras are reduced to states on IMTL-algebras. Furthermore, the necessary and sufficient conditions for finite linearly ordered locally finite MTL-algebras and peculiar MTL-algebras having Bosbach states and Rie?an states are obtained, respectively. In addition, the notions of pseudo-quasi-equivalent and a subalgebra under pseudo-quasi-equivalent are proposed and some of their properties are investigated.     

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.     

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.     

Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded non-commutative residuated lattices

We illustrate by classes of examples the close connections existing between pseudo-MV algebras, on the one hand, and pseudo-BL algebras and divisible bounded non-commutative residuated lattices, on the other hand. We use equivalent definitions of these algebras, as particular cases of pseudo-BCK algebras. We analyse the strongness, the pseudo-involutive center and the filters for each example.     

Pseudo-BCK algebras as partial algebras

It is well-known that the representation of several classes of residuated lattices involves lattice-ordered groups. An often applicable method to determine the representing group (or groups) from a residuated lattice is based on partial algebras: the monoidal operation is restricted to those pairs that fulfil a certain extremality condition, and else left undefined. The subsequent construction applied to the partial algebra is easy, transparent, and leads directly to the structure needed for representation.In this paper, we consider subreducts of residuated lattices, the monoidal and the meet operation being dropped: the resulting algebras are pseudo-BCK semilattices. Assuming divisibility, we can pass on to partial algebras also in this case. To reconstruct the underlying group structure from this partial algebra, if applicable, is again straightforward. We demonstrate the elegance of this method for two classes of pseudo-BCK semilattices: semilinear divisible pseudo-BCK algebras and cone algebras.     

Boolean lifting property for residuated lattices

In this paper we define the Boolean lifting property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP. BLP behaves interestingly in direct products and involutive residuated lattices, and it is closely related to arithmetic properties involving Boolean elements, nilpotent elements and elements of the radical. When BLP is present, strong representation theorems for semilocal and maximal residuated lattices hold.     

Are basic algebras residuated structures?

MV-algebras are bounded commutative integral residuated lattices satisfying the double negation and the divisibility laws. Basic algebras were introduced as a certain generalization of MV-algebras (where associativity and commutativity of the binary operation is neglected). Hence, there is a natural question if also basic algebras can be considered as residuated lattices. We prove that for commutative basic algebras it is the case and for non-commutative ones we involve a modified adjointness condition which gives rise a new generalization of a residuated lattice.     

NMG-代数中同态核的结构刻画

逻辑代数上的Bosbach态与Riečan态是经典概率论中Kolmogorov公理的两种不同方式的多值化推广,也是概率计量逻辑中语义计量化方法的代数公理化,是非经典数理逻辑领域中的重要研究分支.现已证明具有Glivenko性质的逻辑代数上的Bosbach态与Riečan态等价,并且逻辑代数的Glivenko性质是研究态算子的构造和存在性的重要工具,因而是态理论中的研究热点之一.研究了NMG-代数基于核算子的Glivenko性质,证明NMG-代数具有核基Glivenko性质的充要条件是该核算子是从此NMG-代数到其像集代数的同态,并给出NMG-代数中同态核的结构刻画.这里,NMG-代数是刻画序和三角模<（[0,1/2,T_NM]）,（[1/2,1,T_M]）>的逻辑系统NMG的语义逻辑代数.     

Filters of residuated lattices and triangle algebras

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters.In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections. Triangle algebras characterize interval-valued residuated lattices.     

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.     

Bosbach states on fuzzy structures

Pseudo-BL algebras are non-commutative fuzzy structures which generalize BL-algebras and pseudo-MV algebras. In this paper we study the states on a pseudo-BL algebra. This concept is obtained by using the Bosbach condition for each of the two implications of a pseudo BL-algebra. We also propose a notion of conditional state for BL-algebras.The author would like to thank Laureiu Leutean for his valuable suggestions in obtaining the final version of the paper.     

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.     

Very true operators in effect algebras

We introduce the concept of very true operator on an effect algebra. Although an effect algebra is only partial, we define it in the way which is in accordance with traditional definitions in residuated lattices or basic algebras. This is possible if we require monotonicity as an additional condition. We prove that very true operators on effect algebras can be characterized by means of certain subsets which are conditionally complete.     

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. Project supported by the NSF of China (No. 10771524).     

Residuated lattices of block relations: size reduction of concept lattices

We deal with size reduction of concept lattices by means of factorization by block relations defined on corresponding formal context. We show that all block relations with a multiplication defined by means of relational composition form a (non-commutative) residuated lattice. Such residuated lattice can be then thought of as a scale of truth degrees using which we evaluate formulas of predicate logic specifying the desired parameters of the factorization. We also introduce efficient algorithms computing operations on a residuated lattice of block relations. The naive way how to design such algorithms is to compute all the formal concepts of a given context in advance, and then apply some well-known properties of residuated lattices. Our algorithms get rid of the time-consuming precomputation of all concepts.     

Lattice pseudoeffect algebras as double residuated structures

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.     

Extremal states on bounded residuated \ell-monoids with general comparability

Bounded residuated lattice ordered monoids (RlR\ell-monoids) are a common generalization of pseudo-BLBL-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. We investigate bounded RlR\ell-monoids satisfying the general comparability condition in connection with their states (analogues of probability measures). It is shown that if an extremal state on Boolean elements fulfils a simple condition, then it can be uniquely extended to an extremal state on the RlR\ell-monoid, and that if every extremal state satisfies this condition, then the RlR\ell-monoid is a pseudo-BLBL-algebra.     

Indecomposability of free algebras in some subvarieties of residuated lattices and their bounded subreducts

In this paper, we show that free algebras in the variety of residuated lattices and some of its subvarieties are directly indecomposable and show, as a consequence, the direct indecomposability of free algebras for some classes of their bounded implicative subreducts.     

Residuated bilattices

We introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lattice factors. Finally, we show how to employ our product construction to define first-order definable classes of bilattices corresponding to any first-order definable subclass of residuated lattices.