1,456 research outputs found
Soft p-ideals of soft BCI-algebras
AbstractMolodtsov [D. Molodtsov, Soft set theory–First results, Comput. Math. Appl. 37 (1999) 19–31] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Jun [Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413] applied first the notion of soft sets by Molodtsov to the theory of BCK/BCI-algebras. In this paper we introduce the notion of soft p-ideals and p-idealistic soft BCI-algebras, and then investigate their basic properties. Using soft sets, we give characterizations of (fuzzy) p-ideals in BCI-algebras. We provide relations between fuzzy p-ideals and p-idealistic soft BCI-algebras
An analysis of the logic of Riesz Spaces with strong unit
We study \L ukasiewicz logic enriched with a scalar multiplication with
scalars taken in . Its algebraic models, called {\em Riesz MV-algebras},
are, up to isomorphism, unit intervals of Riesz spaces with a strong unit
endowed with an appropriate structure. When only rational scalars are
considered, one gets the class of {\em DMV-algebras} and a corresponding
logical system. Our research follows two objectives. The first one is to deepen
the connections between functional analysis and the logic of Riesz MV-algebras.
The second one is to study the finitely presented MV-algebras, DMV-algebras and
Riesz MV-algebras, connecting them from logical, algebraic and geometric
perspective
The Reticulation of a Universal Algebra
The reticulation of an algebra is a bounded distributive lattice whose prime spectrum of filters or ideals is homeomorphic to the prime
spectrum of congruences of , endowed with the Stone topologies. We have
obtained a construction for the reticulation of any algebra from a
semi-degenerate congruence-modular variety in the case when the
commutator of , applied to compact congruences of , produces compact
congruences, in particular when has principal commutators;
furthermore, it turns out that weaker conditions than the fact that belongs
to a congruence-modular variety are sufficient for to have a reticulation.
This construction generalizes the reticulation of a commutative unitary ring,
as well as that of a residuated lattice, which in turn generalizes the
reticulation of a BL-algebra and that of an MV-algebra. The purpose of
constructing the reticulation for the algebras from is that of
transferring algebraic and topological properties between the variety of
bounded distributive lattices and , and a reticulation functor is
particularily useful for this transfer. We have defined and studied a
reticulation functor for our construction of the reticulation in this context
of universal algebra.Comment: 29 page
Interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras
We introduce the concept of quasi-coincidence of a fuzzy interval value with
an interval valued fuzzy set. By using this new idea, we introduce the notions
of interval valued (\in,\ivq)-fuzzy filters of pseudo -algebras and
investigate some of their related properties. Some characterization theorems of
these generalized interval valued fuzzy filters are derived. The relationship
among these generalized interval valued fuzzy filters of pseudo -algebras
is considered. Finally, we consider the concept of implication-based interval
valued fuzzy implicative filters of pseudo -algebras, in particular, the
implication operators in Lukasiewicz system of continuous-valued logic are
discussed
Tensor products and regularity properties of Cuntz semigroups
The Cuntz semigroup of a C*-algebra is an important invariant in the
structure and classification theory of C*-algebras. It captures more
information than K-theory but is often more delicate to handle. We
systematically study the lattice and category theoretic aspects of Cuntz
semigroups.
Given a C*-algebra , its (concrete) Cuntz semigroup is an object
in the category of (abstract) Cuntz semigroups, as introduced by Coward,
Elliott and Ivanescu. To clarify the distinction between concrete and abstract
Cuntz semigroups, we will call the latter -semigroups.
We establish the existence of tensor products in the category and study
the basic properties of this construction. We show that is a symmetric,
monoidal category and relate with for
certain classes of C*-algebras.
As a main tool for our approach we introduce the category of
pre-completed Cuntz semigroups. We show that is a full, reflective
subcategory of . One can then easily deduce properties of from
respective properties of , e.g. the existence of tensor products and
inductive limits. The advantage is that constructions in are much easier
since the objects are purely algebraic.
We also develop a theory of -semirings and their semimodules. The Cuntz
semigroup of a strongly self-absorbing C*-algebra has a natural product giving
it the structure of a -semiring. We give explicit characterizations of
-semimodules over such -semirings. For instance, we show that a
-semigroup tensorially absorbs the -semiring of the Jiang-Su
algebra if and only if is almost unperforated and almost divisible, thus
establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results
corrected, in particular added 5.2.3-5.2.
Notes on divisible MV-algebras
In these notes we study the class of divisible MV-algebras inside the
algebraic hierarchy of MV-algebras with product. We connect divisible
MV-algebras with -vector lattices, we present the divisible hull as
a categorical adjunction and we prove a duality between finitely presented
algebras and rational polyhedra
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