46 research outputs found
Functorial Properties of the Reticulation of a Universal Algebra
The reticulation of an algebra A is a bounded distributive lattice whose
prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic
to the prime spectrum of congruences of A, with its own Stone topology.
The reticulation allows algebraic and topological properties to be transferred
between the algebra A and this bounded distributive lattice, a transfer which
is facilitated if we can define a reticulation functor from a variety containing A
to the variety of (bounded) distributive lattices. In this paper, we continue the
study of the reticulation of a universal algebra initiated in [27], where we have
used the notion of prime congruence introduced through the term condition
commutator, for the purpose of creating a common setting for the study of the
reticulation, applicable both to classical algebraic structures and to the algebras
of logics. We characterize morphisms which admit an image through th
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
Boolean Lifting Properties for Bounded Distributive Lattices
In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congruences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting properties
A view of canonical extension
This is a short survey illustrating some of the essential aspects of the
theory of canonical extensions. In addition some topological results about
canonical extensions of lattices with additional operations in finitely
generated varieties are given. In particular, they are doubly algebraic
lattices and their interval topologies agree with their double Scott topologies
and make them Priestley topological algebras.Comment: 24 pages, 2 figures. Presented at the Eighth International Tbilisi
Symposium on Language, Logic and Computation Bakuriani, Georgia, September
21-25 200