663 research outputs found
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
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
Algebraic and Topological Results on Lifting Properties in Residuated Lattices
We define lifting properties for universal algebras, which we study in this
general context and then particularize to various such properties in certain
classes of algebras. Next we focus on residuated lattices, in which we
investigate lifting properties for Boolean and idempotent elements modulo
arbitrary, as well as specific kinds of filters. We give topological
characterizations to the lifting property for Boolean elements and several
properties related to it, many of which we obtain by means of the reticulation.Comment: 28 page
-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
The hull-kernel topology on residuated lattices
The notion of hull-kernel topology on a collection of prime filters in a
residuated lattice is introduced and investigated. It is observed that any
collection of prime filters is a topological space under the hull-kernel
and the dual hull-kernel topologies. It is proved that any collection of prime
filters is a space if and only if it is an antichain, and it is a
Hausdorff space if and only if it satisfies a certain condition. Some
characterizations in which maximal filters forms a Hausdorff space are given.
At the end, it is focused on the space of minimal prim filters, and it is shown
that this space is a totally disconnected Hausdorff space. This paper is closed
by a discussion abut the various forms of compactness and connectedness of this
space
Kites and Residuated Lattices
We investigate a construction of an integral residuated lattice starting from
an integral residuated lattice and two sets with an injective mapping from one
set into the second one. The resulting algebra has a shape of a Chinese cascade
kite, therefore, we call this algebra simply a kite. We describe subdirectly
irreducible kites and we classify them. We show that the variety of integral
residuated lattices generated by kites is generated by all finite-dimensional
kites. In particular, we describe some homomorphisms among kites
General coupled semirings of residuated lattices
Di Nola and Gerla showed that MV-algebras and coupled semirings are in a
natural one-to-one correspondence. We generalize this correspondence to
residuated lattices satisfying the double negation law
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
The Conrad Program: From l-groups to algebras of logic
A number of research articles have established the significant role of
lattice-ordered groups (l-groups) in logic. The purpose of the present article
is to lay the groundwork for, and provide significant initial contributions to,
the development of a Conrad type approach to the study of algebras of logic.
The term Conrad Program refers to Paul Conrad's approach to the study of
l-groups, which analyzes the structure of individual l-groups or classes of
l-groups by primarily using strictly lattice theoretic properties of their
lattices of convex l-subgroups. The present article demonstrates that large
parts of the Conrad Program can be profitably extended in the setting of
e-cyclic residuated lattices. An indirect benefit of this work is the
introduction of new tools and techniques in the study of algebras of logic, and
the enhanced role of the lattice of convex subalgebras of a residuated lattice
Generalized Bosbach States
Bosbach states represent a way of probabilisticly evaluating the formulas
from various (commutative or non-commutative) many-valued logics. They are
defined on the algebras corresponding to these logics with values in .
Starting from the observation that in the definition of Bosbach states there
intervenes the standard MV-algebra structure of , in this paper we
introduce Bosbach states defined on residuated lattices with values in
residuated lattices. We are led to two types of generalized Bosbach states,
with distinct behaviours. The properties of generalized Bosbach states, proven
in the paper, may serve as an algebraic foundation for developping some
probabilistic many-valued logics.
{\bf Keywords}: Bosbach states, residuated lattices, MV-algebras, -Cauchy
completion, metric completion.
{\bf MSC 2010}: Primary 06F35. Secondary 06D35.Comment: 27 page
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