35 research outputs found
Study of Pseudo BL–Algebras in View of Left Boolean Lifting Property
In this paper, we define left Boolean lifting property (right Boolean lifting property) LBLP (RBLP) for pseudo BL–algebra which is the property that all Boolean elements can be lifted modulo every left filter (right filter) and next, we study pseudo BL-algebra with LBLP (RBLP). We show that Quasi local, local and hyper Archimedean pseudo BL–algebra that have LBLP (RBLP) has an interesting behavior in direct products. LBLP (RBLP) provides an important representation theorem for semi local and maximal pseudo BL–algebra
Lattice-theoretic properties of algebras of logic
Abstract In the theory of lattice-ordered groups, there are interesting examples of properties -such as projectability -that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semilinear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a, 1] is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety
Scheme theoretic tropicalization
In this paper, we introduce ordered blueprints and ordered blue schemes,
which serve as a common language for the different approaches to
tropicalizations and which enhances tropical varieties with a schematic
structure. As an abstract concept, we consider a tropicalization as a moduli
problem about extensions of a given valuation between ordered
blueprints and . If is idempotent, then we show that a
generalization of the Giansiracusa bend relation leads to a representing object
for the tropicalization, and that it has yet another interpretation in terms of
a base change along . We call such a representing object a scheme theoretic
tropicalization.
This theory recovers and improves other approaches to tropicalizations as we
explain with care in the second part of this text.
The Berkovich analytification and the Kajiwara-Payne tropicalization appear
as rational point sets of a scheme theoretic tropicalization. The same holds
true for its generalization by Foster and Ranganathan to higher rank
valuations.
The scheme theoretic Giansiracusa tropicalization can be recovered from the
scheme theoretic tropicalizations in our sense. We obtain an improvement due to
the resulting blueprint structure, which is sufficient to remember the
Maclagan-Rinc\'on weights.
The Macpherson analytification has an interpretation in terms of a scheme
theoretic tropicalization, and we give an alternative approach to Macpherson's
construction of tropicalizations.
The Thuillier analytification and the Ulirsch tropicalization are rational
point sets of a scheme theoretic tropicalization. Our approach yields a
generalization to any, possibly nontrivial, valuation with
idempotent and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the
paper, please cf. the paragraph "Differences to previous versions" in the
introductio
A uniform classification of discrete series representations of affine Hecke algebras
We give a new and independent parameterization of the set of discrete series
characters of an affine Hecke algebra , in terms of a
canonically defined basis of a certain lattice of virtual
elliptic characters of the underlying (extended) affine Weyl group. This
classification applies to all semisimple affine Hecke algebras ,
and to all , where denotes the vector
group of positive real (possibly unequal) Hecke parameters for .
By analytic Dirac induction we define for each a
continuous (in the sense of [OS2]) family
,
such that (for
some ) is an irreducible discrete series
character of . Here
is a finite union of hyperplanes in
.
In the non-simply laced cases we show that the families of virtual discrete
series characters are piecewise rational
in the parameters . Remarkably, the formal degree of
in such piecewise rational family turns
out to be rational. This implies that for each there
exists a universal rational constant determining the formal degree in the
family of discrete series characters
. We will compute
the canonical constants , and the signs . For
certain geometric parameters we will provide the comparison with the
Kazhdan-Lusztig-Langlands classification.Comment: 31 pages, 2 table
Orderings and Boolean algebras not isomorphic to recursive ones
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1967.Vita.Includes bibliographies.Lawrence Feiner.Ph.D
Fuzzy Mathematics
This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value
Advances and Applications of Dezert-Smarandache Theory (DSmT) for Information Fusion (Collected works), Vol. 2
This second volume dedicated to Dezert-Smarandache Theory (DSmT) in Information Fusion brings in new fusion quantitative rules (such as the PCR1-6, where PCR5 for two sources does the most mathematically exact redistribution of conflicting masses to the non-empty sets in the fusion literature), qualitative fusion rules, and the Belief Conditioning Rule (BCR) which is different from the classical conditioning rule used by the fusion community working with the Mathematical Theory of Evidence.
Other fusion rules are constructed based on T-norm and T-conorm (hence using fuzzy logic and fuzzy set in information fusion), or more general fusion rules based on N-norm and N-conorm (hence using neutrosophic logic and neutrosophic set in information fusion), and an attempt to unify the fusion rules and fusion theories.
The known fusion rules are extended from the power set to the hyper-power set and comparison between rules are made on many examples.
One defines the degree of intersection of two sets, degree of union of two sets, and degree of inclusion of two sets which all help in improving the all existing fusion rules as well as the credibility, plausibility, and communality functions.
The book chapters are written by Frederic Dambreville, Milan Daniel, Jean Dezert, Pascal Djiknavorian, Dominic Grenier, Xinhan Huang, Pavlina Dimitrova Konstantinova, Xinde Li, Arnaud Martin, Christophe Osswald, Andrew Schumann, Tzvetan Atanasov Semerdjiev, Florentin Smarandache, Albena Tchamova, and Min Wang