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.     

Commutative bounded integral residuated orthomodular lattices are Boolean algebras

We show that a commutative bounded integral orthomodular lattice is residuated iff it is a Boolean algebra. This result is a consequence of (Ward, Dilworth in Trans Am Math Soc 45, 336–354, 1939, Theorem 7.31); however, out proof is independent and uses other instruments.     

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).     

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.     

Free algebras in varieties of Stonean residuated lattices

Given a variety of bounded residuated lattices satisfying the Stone identity , the free algebras in over a set X of cardinality |X| are represented as weak Boolean products over the Cantor space 2^|X| of a family of free algebras in an associated variety of (not necessarily bounded) residuated lattices with a bottom added.     

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.     

Fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices

This paper presents basic notions about fuzzy measures over algebras of fuzzy subsets of a fuzzy set. It also presents basic ideas on fuzzy integrals defined using these fuzzy measures. Definitions of new types of fuzzy measures and integrals are motivated by our research on generalized quantifiers. Several useful properties of fuzzy measures and fuzzy integrals are stated and proved. Definitions presented in this paper and its results will be employed in subsequent papers on generalized quantifiers defined using this type of fuzzy integral.     

Portfolio optimization when risk factors are conditionally varying and heavy tailed

Assumptions about the dynamic and distributional behavior of risk factors are crucial for the construction of optimal portfolios and for risk assessment. Although asset returns are generally characterized by conditionally varying volatilities and fat tails, the normal distribution with constant variance continues to be the standard framework in portfolio management. Here we propose a practical approach to portfolio selection. It takes both the conditionally varying volatility and the fat-tailedness of risk factors explicitly into account, while retaining analytical tractability and ease of implementation. An application to a portfolio of nine German DAX stocks illustrates that the model is strongly favored by the data and that it is practically implementable.     

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.     

On filter theory of residuated lattices

The aim of this paper is to develop the filter theory of general residuated lattices. First, we extend some particular types of filters and fuzzy filters in BL-algebras and MTL-algebras naturally to general residuated lattices, and further enumerate some relative results obtained in BL-algebras or MTL-algebras, which still hold in general residuated lattices. Next, we introduce the concepts of regular filters and fuzzy regular filters to general residuated lattices, which are two new types of filters and fuzzy filters, and derive some of their characterizations. Finally, we discuss the relations between (fuzzy) regular filters and several other special (fuzzy) filters, and also characterize some special classes of residuated lattices by filters or fuzzy filters.     

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).     

Some types of filters in residuated lattices

In this paper, inspired by some types of $BL$ -algebra filters (deductive systems) introduced in Haveshki et al. (Soft Comput 10:657–664, 2006), Kondo and Dudek (Soft Comput 12:419–423, 2008) and Turunen (Arch Math Log 40:467–473, 2001), we defined residuated lattice versions of them and study them in connection with Van Gasse et al. (Inf Sci 180(16):3006–3020, 2010), Lianzhen and Kaitai (Inf Sci 177:5725–5738, 2007), Zhu and Xu (Inf Sci 180:3614–3632, 2010). Also we consider some relations between these filters and quotient residuated lattice that are constructed via these filters.     

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.     

Characterization of extended filters in residuated lattices

We give a characterization theorem of extended filters on residuated lattices, from which many results are immediately obtained. We show that, for a bounded integral commutative residuated lattice X, (1) an extended filter $E_F (B)$ associated with $B$ is characterized by $E_F (B) = [B) \rightarrow F$ , where $B\subseteq X$ and $F$ is a filter of $X$ ; (2) the class $E(B)$ of all extended filters associated with $B$ is a complete Heyting algebra. (3) the class $S(B)$ of all stable filters relative to $B\subseteq X$ is also a complete Heyting algebra.     

Effect algebras with the subsequential interpolation property

We prove Brooks–Jewett, Vitali–Hahn–Saks and NikodymBoundedness theorems for modular measures on lattice-ordered effect algebras with the subsequential interpolation property. Supported by G.N.A.M.P.A.