3 research outputs found
-normal residuated lattices
The notion of -normal residuated lattice, as a class of residuated
lattices in which every prime filter contains at most minimal prime
filters, is introduced and studied. Before that, the notion of -filter
is introduced and it is observed that the set of -filters in a
residuated lattice forms a distributive lattice on its own, which includes the
set of coannulets as a sublattice. The class of -normal residuated lattices
is characterized in terms of their prime filters, minimal prime filters,
coannulets and -filters.Comment: arXiv admin note: text overlap with arXiv:1812.1151
Quasicomplemented residuated lattices
In this paper, the class of quasicomplemented residuated lattices is
introduced and investigated, as a subclass of residuated lattices in which any
prime filter not containing any dense element is a minimal prime filter. The
notion of disjunctive residuated lattices is introduced and it is observed that
a residuated lattice is Boolean if and only if it is disjunctive and
quasicomplemented. Finally, some characterizations for quasicomplemented
residuated lattices are given by means of the new notion of -filters.Comment: arXiv admin note: text overlap with arXiv:1812.11511,
arXiv:1812.1151
Mp-residuated lattices
This paper is devoted to the study of a fascinating class of residuated
lattices, the so-called mp-residuated lattice, in which any prime filter
contains a unique minimal prime filter. A combination of algebraic and
topological methods is applied to obtain new and structural results on
mp-residuated lattices. It is demonstrated that mp-residuated lattices are
strongly tied up with the dual hull-kernel topology. Especially, it is shown
that a residuated lattice is mp if and only if its minimal prime spectrum,
equipped with the dual hull-kernel topology, is Hausdorff if and only if its
prime spectrum, equipped with the dual hull-kernel topology, is normal. The
class of mp-residuated lattices is characterized by means of pure filters. It
is shown that a residuated lattice is mp if and only if its pure filters are
precisely its minimal prime filters, if and only if its pure spectrum is
homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel
topology.Comment: 26 pages, 2 figures. arXiv admin note: substantial text overlap with
arXiv:2202.1011