scholarly journals Representation of $L$-fuzzy binary relations via a Galois connection

2009 ◽  
Vol 40 (3) ◽  
pp. 287-305
Author(s):  
Nistala V. E. S. Murthy ◽  
Peruru G. Prasad
Keyword(s):  
Complete Lattice ◽  
Galois Connection ◽  
Binary Relations ◽  
Truth Values ◽  
Galois Connections ◽  
Fuzzy Point ◽  
Point Set

Our aim in this Paper is to establish Galois connections between various types of fuzzy binary relations and fuzzy I-ary relations on a crisp set, that take their truth values in a complete lattice, and same type of crisp binary and I-ary relations on the associated fuzzy-point-set.

scholarly journals A bird’s eye view of the connections among L-fuzzy i-covers, c-covers and (L-fuzzy) tolerances of L-fuzzy (point) set.

2021 ◽  
Vol 1767 (1) ◽  
pp. 012052
Author(s):  
G Sai Prasanthi ◽  
Pusuluri V N H Ravi ◽  
Nistala V E S Murthy
Keyword(s):  
Fuzzy Point ◽  
Point Set

scholarly journals ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

2010 ◽  
Vol 83 (2) ◽  
pp. 273-288 ◽  
Author(s):  
D. G. FITZGERALD ◽  
KWOK WAI LAU
Keyword(s):  
Transformation Semigroup ◽  
Galois Connection ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Binary Relations ◽  
Ideal Lattices ◽  
New Interpretation ◽  
Semigroup Of Transformations ◽  
The Ideal

AbstractThe partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

scholarly journals Lattices of fuzzy objects

1996 ◽  
Vol 19 (4) ◽  
pp. 759-766 ◽  
Author(s):  
Arturo A. L. Sangalli
Keyword(s):  
Complete Lattice ◽  
Binary Relations ◽  
Special Cases ◽  
Fuzzy Objects ◽  
Fuzzy Subgroups ◽  
Special Case ◽  
Fuzzy Subsets

The collection of fuzzy subsets of a setXforms a complete lattice that extends the complete lattice𝒫(X)of crisp subsets ofX. In this paper, we interpret this extension as a special case of the “fuzzification” of an arbitrary complete latticeA. We show how to construct a complete latticeF(A,L)–theL-fuzzificatio ofA, whereLis the valuation lattice– that extendsAwhile preserving all suprema and infima. The “fuzzy” objects inF(A,L)may be interpreted as the sup-preserving maps fromAto the dual ofL. In particular, each complete lattice coincides with its2-fuzzification, where2is the twoelement lattice. Some familiar fuzzifications (fuzzy subgroups, fuzzy subalgebras, fuzzy topologies, etc.) are special cases of our construction. Finally, we show that the binary relations on a setXmay be seen as the fuzzy subsets ofXwith respect to the valuation lattice𝒫(X).

On an algebra of lattice-valued logic

2005 ◽  
Vol 70 (1) ◽  
pp. 282-318
Author(s):  
Lars Hansen
Keyword(s):  
Boolean Algebra ◽  
Complete Lattice ◽  
Algebraic Theory ◽  
Propositional Calculus ◽  
Boolean Lattice ◽  
Finite Chain ◽  
Axiomatic Theory ◽  
Truth Values ◽  
Algebraic Generalization

AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.

scholarly journals The lattice and semigroup structure of multipermutations

2021 ◽  
pp. 1-25
Author(s):  
Catarina Carvalho ◽  
Barnaby Martin
Keyword(s):  
Lattice Structure ◽  
Galois Connection ◽  
Order Logic ◽  
First Order Logic ◽  
Binary Relations ◽  
Regular Elements ◽  
First Order ◽  
Vertex Set ◽  
Composition Of Relations ◽  
Text Element

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an [Formula: see text]-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in [Formula: see text]or to be [Formula: see text]-complete. We go on to study the monoid of all multipermutations on an [Formula: see text]-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on [Formula: see text].

scholarly journals Galois connection of stabilizers in residuated lattices

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1223-1239
Author(s):  
Saeed Rasouli
Keyword(s):  
Complete Lattice ◽  
Residuated Lattice ◽  
Galois Connection ◽  
Residuated Lattices ◽  
Pseudocomplemented Lattices

The paper is devoted to introduce the notions of some types of stabilizers in non-commutative residuated lattices and to investigate their properties. We establish a connection between (contravariant) Galois connection and stabilizers of a residuated lattices. If A is a residuated lattice and F be a filter of A, we show that the set of all stabilizers relative to F of a same type forms a complete lattice. Furthermore, we prove that ST - F?l, ST - Fl and ST - Fs are pseudocomplemented lattices.

scholarly journals Short Note on Topological Fuzzy Spaces of the Behaviour of GS fuzzy Closed Sets

2018 ◽  
Vol 7 (4.10) ◽  
pp. 810
Author(s):  
M. Mary Victoria Florence ◽  
E. Priyadarshini ◽  
M. Vidhya ◽  
A. Govindarajan ◽  
E. P.Siva
Keyword(s):  
Short Note ◽  
Good Interest ◽  
Topological Spaces ◽  
Closed Sets ◽  
Fuzzy Point ◽  
Point Set ◽  
Fuzzy Topological Spaces

Many fuzzy topologists have very good interest in generalized fuzzy closed sets and in fuzzy point set topology. Here, properties of GS  in fuzzy topological spaces and its relationship with other generalized fuzzy closed sets has been discussed. 

A RING-THEORETIC BASIS FOR LOGIC PROGRAMMING

1990 ◽  
Vol 01 (01) ◽  
pp. 23-48 ◽  
Author(s):  
V.S. SUBRAHMANIAN
Keyword(s):  
Logic Programming ◽  
Algebraic Structure ◽  
Complete Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Truth Values ◽  
Partially Ordered

Investigations into the semantics of logic programming have largely restricted themselves to the case when the set of truth values being considered is a complete lattice. While a few theorems have been obtained which make do with weaker structures, to our knowledge there is no semantical characterization of logic programming which does not require that the set of truth values be partially ordered. We derive here semantical results on logic programming over a space of truth values that forms a commutative pseudo-ring (an algebraic structure a bit weaker than a ring) with identity. This permits us to study the semantics of multi-valued logic programming having a (possibly) non-partially ordered set of truth values.

scholarly journals Galois connections between sets of paths and closure operators in simple graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 1573-1581 ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Positive Integer ◽  
Digital Images ◽  
Galois Connection ◽  
Simple Graph ◽  
Closure Operators ◽  
Simple Graphs ◽  
Galois Connections ◽  
Digital Plane ◽  
Vertex Set

AbstractFor every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

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