1,415 research outputs found
Boolean Lifting Property for Residuated Lattices
In this paper we define the Boolean Lifting Property (BLP) for residuated
lattices to be the property that all Boolean elements can be lifted modulo
every filter, and study residuated lattices with BLP. Boolean algebras, chains,
local and hyperarchimedean residuated lattices have BLP. BLP behaves
interestingly in direct products and involutive residuated lattices, and it is
closely related to arithmetic properties involving Boolean elements, nilpotent
elements and elements of the radical. When BLP is present, strong
representation theorems for semilocal and maximal residuated lattices hold.Comment: 28 page
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
Relatively residuated lattices and posets
It is known that every relatively pseudocomplemented lattice is residuated
and, moreover, it is distributive. Unfortunately, non-distributive lattices
with a unary operation satisfying properties similar to relative
pseudocomplementation cannot be converted in residuated ones. The aim of our
paper is to introduce a more general concept of a relative residuated lattice
in such a way that also non-modular sectionally pseudocomplemented lattices are
included. We derive several properties of relative residuated lattices which
are similar to those known for residuated ones and extend our results to
posets
Integrally Closed Residuated Lattices
A residuated lattice is defined to be integrally closed if it satisfies the
equations x\x = e and x/x = e. Every integral, cancellative, or divisible
residuated lattice is integrally closed, and, conversely, every bounded
integrally closed residuated lattice is integral. It is proved that the mapping
a -> (a\e)\e on any integrally closed residuated lattice is a homomorphism onto
a lattice-ordered group. A Glivenko-style property is then established for
varieties of integrally closed residuated lattices with respect to varieties of
lattice-ordered groups, showing in particular that integrally closed residuated
lattices form the largest variety of residuated lattices admitting this
property with respect to lattice-ordered groups. The Glivenko property is used
to obtain a sequent calculus admitting cut-elimination for the variety of
integrally closed residuated lattices and to establish the decidability, indeed
PSPACE-completenes, of its equational theory. Finally, these results are
related to previous work on (pseudo) BCI-algebras, semi-integral residuated
partially ordered monoids, and algebras for Casari's comparative logic
Involutive right-residuated l-groupoids
A common generalization of orthomodular lattices and residuated lattices is
provided corresponding to bounded lattices with an involution and sectionally
extensive mappings. It turns out that such a generalization can be based on
integral right-residuated l-groupoids. This general framework is applied to
MV-algebras,orthomodular lattices, Nelson algebras, basic algebras and Heyting
algebras.Comment: 22 page
n-fold filters in residuated lattice
Residuated lattices play an important role in the study of fuzzy logic based
of t-norm. In this paper, we introduced the notions of n-fold implicative
filters, n-fold positive implicative filters, n-fold boolean filters, n-fold
fantastic filters, n-fold normal filters and n-fold obstinate filters in
residuated lattices and study the relations among them. This generalized the
similar existing results in BL-algebra with the connection of the work of Kerre
and all in [14], Kondo and all in [7], [11] and Motamed and all in [9]. At the
end of this paper, we draw two diagrams; the first one describe the relations
between some type of n-fold filters in residuated lattices and the second one
describe the relations between some type of n-fold residuated lattices
Nilpotency and the Hamiltonian property for cancellative residuated lattices
The present article studies nilpotent and Hamiltonian cancellative residuated
lattices and their relationship with nilpotent and Hamiltonian lattice-ordered
groups. In particular, results about lattice-ordered groups are extended to the
domain of residuated lattices. The two key ingredients that underlie the
considerations of this paper are the categorical equivalence between Ore
residuated lattices and lattice-ordered groups endowed with a suitable modal
operator; and Malcev's description of nilpotent groups of a given nilpotency
class c in terms of a semigroup equation
-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
A categorical equivalence for Stonean residuated lattices
Distributive Stonean residuated lattices are closely related to Stone
algebras since their bounded lattice reduct is a Stone algebra. In the present
work we follow the ideas presented by Chen and Gr\"{a}tzer and try to apply
them for the case of Stonean residuated lattices. Given a Stonean residuated
lattice, we consider the triple formed by its Boolean skeleton, its algebra of
dense elements and a connecting map. We define a category whose objects are
these triples and suitably defined morphisms, and prove that we have a
categorical equivalence between this category and that of Stonean residuated
lattices. We compare our results with other works and show some applications of
the equivalence
When does a semiring become a residuated lattice?
It is an easy observation that every residuated lattice is in fact a semiring
because multiplication distributes over join and the other axioms of a semiring
are satisfied trivially. This semiring is commutative, idempotent and simple.
The natural question arises if the converse assertion is also true. We show
that the conversion is possible provided the given semiring is, moreover,
completely distributive. We characterize semirings associated to complete
residuated lattices satisfying the double negation law where the assumption of
complete distributivity can be omitted. A similar result is obtained for
idempotent residuated lattices
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