90 research outputs found
A note on distributivity of the lattice of L-ideals of a ring
Many studies have investigated the lattice of fuzzy substructures of algebraic structures such as groups and rings. In this study, we prove that the lattice of L-ideals of a ring is distributive if and only if the lattice of its ideals is distributive, for an infinitely ?- distributive lattice L. © 2019 Hacettepe University. All rights reserved
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Non-additive probabilities and quantum logic in finite quantum systems
YesA quantum system Σ(d) with variables in Z(d) and with Hilbert space H(d), is considered. It is shown that the additivity relation of Kolmogorov probabilities, is not valid in the Birkhoff-von Neumann orthocomplemented modular lattice of subspaces L(d). A second lattice Λ(d) which is distributive and contains the subsystems of Σ(d) is also considered. It is shown that in this case also, the additivity relation of Kolmogorov probabilities is not valid. This suggests that a more general (than Kolmogorov) probability theory is needed, and here we adopt the Dempster-Shafer probability theory. In both of these lattices, there are sublattices which are Boolean algebras, and within these 'islands' quantum probabilities are additive
Pseudocomplementation in (normal) subgroup lattices
The goal of this article is to study finite groups admitting a pseudocomplemented subgroup lattice (PK-groups) or a pseudocomplemented normal subgroup lattice (PKN-groups). In particular, we obtain a complete classification of finite PK-groups and of finite nilpotent PKN-groups. We also study groups with a Stone normal subgroup lattice, and we classify finite groups for which every subgroup has a Stone normal subgroup lattice. Finally, we obtain a complete classification of finite groups for which every subgroup is monolithic
Möbius operators and non-additive quantum probabilities in the Birkhoff-von Neumann lattice.
yesThe properties of quantum probabilities are linked to the geometry of quantum mechanics, described
by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property
of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Möbius (or non-additivity) operators which are defined through Möbius transforms, and which are shown to be intimately related to commutators.
The lack of distributivity in the Birkhoff-von Neumann lattice Λd, causes deviations from the law of the total probability (which is central in Kolmogorov’s probability theory). Projectors which quantify the lack of distributivity in Λd, and also deviations from the law of the total probability, are introduced. All these operators, are observables and they can be measured experimentally. Constraints for the Möbius operators, which are based on the properties of the Birkhoff-von Neumann
lattice (which in the case of finite quantum systems is a modular lattice), are derived. Application of this formalism in the context of coherent states, generalizes coherence to multi-dimensional structures
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
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