31 research outputs found
A Note on Direct Products, Subreducts and Subvarieties of PBZ*--lattices
PBZ*--lattices are bounded lattice--ordered structures arising in the study
of quantum logics, which include orthomodular lattices, as well as
antiortholattices. Antiortholattices turn out not only to be directly
irreducible, but also to have directly irreducible lattice reducts. Their
presence in varieties of PBZ*--lattices determines the lengths of the subposets
of dense elements of the members of those varieties. The variety they generate
includes two disjoint infinite ascending chains of subvarieties, and the
lattice of subvarieties of the variety of pseudo--Kleene algebras can be
embedded as a poset in the lattice of subvarieties of its subvariety formed of
its members that satisfy the Strong De Morgan condition. We obtain
axiomatizations for all members of a complete sublattice of the lattice of
subvarieties of this latter variety axiomatized by the Strong De Morgan
identity with respect to the variety generated by antiortholattices.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