118 research outputs found
Transferring Davey`s Theorem on Annihilators in Bounded Distributive Lattices to Modular Congruence Lattices and Rings
Congruence lattices of semiprime algebras from semi--degenerate
congruence--modular varieties fulfill the equivalences from B. A. Davey`s
well--known characterization theorem for --Stone bounded distributive
lattices, moreover, changing the cardinalities in those equivalent conditions
does not change their validity. I prove this by transferring Davey`s Theorem
from bounded distributive lattices to such congruence lattices through a
certain lattice morphism and using the fact that the codomain of that morphism
is a frame. Furthermore, these equivalent conditions are preserved by finite
direct products of such algebras, and similar equivalences are fulfilled by the
elements of semiprime commutative unitary rings and, dualized, by the elements
of complete residuated lattices.Comment: 18 page
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
Priestley duality for MV-algebras and beyond
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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