Belief Functions on MV-Algebras of Fuzzy Events Based on Fuzzy Evidence

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

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Extended two-dimensional belief function based on divergence measurement

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201727 ◽

2020 ◽

pp. 1-8

Author(s):

Jianping Fan ◽

Jing Wang ◽

Meiqin Wu

Keyword(s):

Belief Function ◽

Distribution Functions ◽

Divergence Measure ◽

Uncertain Information ◽

Belief Functions ◽

Two Dimensional ◽

Probability Distribution Functions ◽

Basic Probability ◽

The Difference ◽

Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

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Priestley duality for MV-algebras and beyond

Forum Mathematicum ◽

10.1515/forum-2020-0115 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wesley Fussner ◽

Mai Gehrke ◽

Samuel J. van Gool ◽

Vincenzo Marra

Keyword(s):

Large Class ◽

Order Conditions ◽

Enriched Environment ◽

Priestley Duality ◽

Distributive Lattices ◽

First Order ◽

Dual Spaces ◽

Binary Operations ◽

New Perspective ◽

Mv Algebras

Abstract We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

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