## An example of L-fuzzy Join Space (2001)

Venue: | Rend. Circ. Mat. Palermo |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Kehagias01anexample,

author = {Ath. Kehagias},

title = {An example of L-fuzzy Join Space},

journal = {Rend. Circ. Mat. Palermo},

year = {2001},

volume = {51},

pages = {503--526}

}

### OpenURL

### Abstract

On a generalized deMorgan lattice (X, _<, V, A,') we introduce a family of join hyperoperations *p, parametrized by a parameter p X. As a result we obtain a family of join spaces (X, ,p). We show that: for every a, b X the family {a*p b}pex can be considered as the p-cuts of a L-fuzzy set a * b; in this manner we synthesize a L-fuzzy hyperoperation which takes pairs from X to L-fuzzy subsets of X. We then show that (X, ,p) is a L-fuzzy hypergroup (in the sense of Corsini) and can be considered as a L-fuzzy join space. Furthermore, a * b is a L-fuzzy interval for all a, b X.

### Citations

587 | Lattice Theory - Birkhoff - 1967 |

191 |
First Course in Fuzzy Logic
- Nguyen, Walker
- 1999
(Show Context)
Citation Context .... for M, NsF(X) we write M _ X we have: M(x) _ For all M, NsF(X): we define the L-fuzzy set MYN by: (MYN)(x) ---- M(x)N(x); we define the L-fuzzy set MAN by: (MAN)(x) ---- M(x)AN(x). It is well known =-=[15]-=- that _ F(X) and that (F(X), _, A) is a complete and distributive lattice with sup(M, N) ---- MsN, inf(M, N) = M A N. 6. For all M,NsP(X) we write M ,- N iff 3xsMf N. For M,NsF(X) and psX we write M ,... |

83 | Lattice Theory - Birkho¤ - 1979 |

16 | Fuzzy Lattice Neurocomputing (FLN) models”, Neural Networks
- Kaburlasos, Petridis
- 2000
(Show Context)
Citation Context ... far as we know, has not been undertaken so far. Last but not least, let us remark that fuzzy lattices and join hyperoperations have been used in several engineering and computer science applications =-=[10, 16, 17]-=-. Hence we expect that the results of this work may in the future prove useful in similar applications. A Appendix: L-Fuzzy Lattices and Intervals We start with some informal remarks. Consider two com... |

15 |
Prolegomena of hypergroup theory
- Corsini
- 1993
(Show Context)
Citation Context ...first briefly review some work on these themes. Hypergroups have been introduced in [14] and studied extensively by many mathematicians. A comprehensive review of the theory of hypergroups appears in =-=[3]-=-. Join spaces have been introduced by Jantosciak and Prenowitz [18]. The topic is covered in considerable detail in [19]. Jantosciak has noted that join spaces are a special case of hypergroups [9]. I... |

15 | Fuzzy lattice neural network (FLNN): a hybrid model for learning
- Petridis, Kaburlasos
- 1998
(Show Context)
Citation Context ... far as we know, has not been undertaken so far. Last but not least, let us remark that fuzzy lattices and join hyperoperations have been used in several engineering and computer science applications =-=[10, 16, 17]-=-. Hence we expect that the results of this work may in the future prove useful in similar applications. A Appendix: L-Fuzzy Lattices and Intervals We start with some informal remarks. Consider two com... |

14 |
Hypergraphs and hypergroups
- Corsini
(Show Context)
Citation Context ... algebraic hyperstructures. Corsini and Zahedi introduced some crisp algebraic hyperstructures related to fuzzy sets [5, 6, 30]. More recently, the concept of fuzzy hypergroups has been introduced in =-=[4]-=-; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied... |

12 | Hyperstructures and their Representations - Vougiouklis - 1994 |

11 |
Learning in the Framework of Fuzzy Lattices
- Petridis, Kaburlasos
- 1999
(Show Context)
Citation Context ... far as we know, has not been undertaken so far. Last but not least, let us remark that fuzzy lattices and join hyperoperations have been used in several engineering and computer science applications =-=[10, 16, 17]-=-. Hence we expect that the results of this work may in the future prove useful in similar applications. A Appendix: L-Fuzzy Lattices and Intervals We start with some informal remarks. Consider two com... |

9 |
Join geometries
- Prenowitz, Jantosciak
- 1979
(Show Context)
Citation Context ...y mathematicians. A comprehensive review of the theory of hypergroups appears in [3]. Join spaces have been introduced by Jantosciak and Prenowitz [18]. The topic is covered in considerable detail in =-=[19]-=-. Jantosciak has noted that join spaces are a special case of hypergroups [9]. In recent years there has been considerable activity on fuzzy algebraic hyperstructures. Corsini and Zahedi introduced so... |

8 | Fuzzy lattices
- Ajmal, Thomas
- 1994
(Show Context)
Citation Context ...ly related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied in more detail in =-=[1]-=-. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice theoretic sense, as a special case of fuzzy lattices) are studied in [13]. In this paper we do the following. On a ... |

8 | On F-polygroups and fuzzy sub-F-polygroups - Hasankhani, Zahedi - 1998 |

8 |
Geometries and Join Spaces
- Prenowitz, Jantosciak
- 1972
(Show Context)
Citation Context ...een introduced in [14] and studied extensively by many mathematicians. A comprehensive review of the theory of hypergroups appears in [3]. Join spaces have been introduced by Jantosciak and Prenowitz =-=[18]-=-. The topic is covered in considerable detail in [19]. Jantosciak has noted that join spaces are a special case of hypergroups [9]. In recent years there has been considerable activity on fuzzy algebr... |

7 | The lattice of fuzzy intervals and sufficient conditions for its distributivity". arXiv:cs.OH/0206025, at http://xxx.lanl.gov/find/cs
- Kehagias
(Show Context)
Citation Context ...uced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice theoretic sense, as a special case of fuzzy lattices) are studied in =-=[13]-=-. In this paper we do the following. On a generalized aleMorgan lattice (X, _, A/) we introduce a family of join hyperoperations .p, parametrized by a parameter psX. As a result we obtain a family of ... |

7 |
The fundamental relations on Hv-rings
- Vougiouklis
- 1994
(Show Context)
Citation Context ...b Op Cp (p-associativity). (ii) For all asX we have aSX = X (R3 reproducibility). We now introduce an additional definition which makes use of Hv associativity (see Spartalis [20, 21] and Vougiouklis =-=[22, 25, 26]-=-). Definition 4.15 Given a hyperoperations: X x X - F(X), the hyperstructure (X, ) is called Lfuzzy$-Hv-hypergroup if it satisfies the following conditions. (i) For all a, b, c,psX we have (asb)sc ..p... |

6 |
Join Spaces, Power Sets, Fuzzy Sets
- Corsini
- 1994
(Show Context)
Citation Context ...e of hypergroups [9]. In recent years there has been considerable activity on fuzzy algebraic hyperstructures. Corsini and Zahedi introduced some crisp algebraic hyperstructures related to fuzzy sets =-=[5, 6, 30]-=-. More recently, the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings... |

6 |
Join Spaces associated with Fuzzy Sets
- Corsini, Leoreanu
- 1995
(Show Context)
Citation Context ...e of hypergroups [9]. In recent years there has been considerable activity on fuzzy algebraic hyperstructures. Corsini and Zahedi introduced some crisp algebraic hyperstructures related to fuzzy sets =-=[5, 6, 30]-=-. More recently, the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings... |

5 |
L-fuzzy lattices: an introduction
- Tepavcevic, Trajkovski
- 2001
(Show Context)
Citation Context ...oduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and =-=[24]-=-. Fuzzy intervals (in a lattice theoretic sense, as a special case of fuzzy lattices) are studied in [13]. In this paper we do the following. On a generalized aleMorgan lattice (X, _, A/) we introduce... |

4 |
On Hv-group rings
- Spartalis, Dramalides, et al.
- 1988
(Show Context)
Citation Context ...ave (a Op b) Opc = a Op (b Op Cp (p-associativity). (ii) For all asX we have aSX = X (R3 reproducibility). We now introduce an additional definition which makes use of Hv associativity (see Spartalis =-=[20, 21]-=- and Vougiouklis [22, 25, 26]). Definition 4.15 Given a hyperoperations: X x X - F(X), the hyperstructure (X, ) is called Lfuzzy$-Hv-hypergroup if it satisfies the following conditions. (i) For all a,... |

4 |
Fuzzy ideals and congruences of lattices
- Swamy, Raju
- 1998
(Show Context)
Citation Context ...been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in =-=[23]-=- and [24]. Fuzzy intervals (in a lattice theoretic sense, as a special case of fuzzy lattices) are studied in [13]. In this paper we do the following. On a generalized aleMorgan lattice (X, _, A/) we ... |

4 | The fundamental relations of H v -rings - Spartalis, Vougiouklis - 1994 |

4 | F -Polygroups - Zahedi, Hasankhani - 1996 |

3 |
F-Hyperrings". Ital
- Hasankhani, Zahedi
- 1998
(Show Context)
Citation Context ... the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in =-=[7]-=-. Fuzzy lattices have been introduced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice theoretic sense, as a special case o... |

3 | Lattice-ordered join space: an applications-oriented example
- Kehagias, Konstantinidou
(Show Context)
Citation Context ...sequel. Proposition 2.2 For all a, b, x, ysX such that x _ (i) aV Ix, Y]: [aV x, aVy]; (ii) aA Ix, Y]: [aAx, aAy]; (iii) [a, b] V Ix, y]: [a V x, b V y]; (iv) [a,b] A Ix, Y]: [a A x,b A y]. Proof. In =-=[11]-=-.sThe following propositions describe some properties of p-cuts. Their proofs can be found in [15]. Proposition 2.3 Take any MsF with p-cuts {Mp}pc X. Then we have the following. (i) For all p, qsX we... |

3 |
F-Polygroups
- Zahedi, Hasankhani
- 1996
(Show Context)
Citation Context ...d to fuzzy sets [5, 6, 30]. More recently, the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in =-=[28, 29]-=-. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice... |

3 |
F-Polygroups (II
- Zahedi, Hasankhani
- 1996
(Show Context)
Citation Context ...d to fuzzy sets [5, 6, 30]. More recently, the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in =-=[28, 29]-=-. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in [27] and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice... |

3 | F -Hyperrings". Ital - Hasankhani, Zahedi - 1998 |

3 | On H V -group rings - Spartalis, Dramalides, et al. - 1998 |

3 | Classical geometries as hypergroups”, in Convegno su Ipergruppi, Altre Strutture Multivoche e loro Applicazioni - Jantosciak - 1985 |

2 |
Classical geometries as hypergroups", in Convegno su Ipergruppi, Altre Strutture Multivoche e loro Applicazioni
- Jantosciak
- 1985
(Show Context)
Citation Context ...in [3]. Join spaces have been introduced by Jantosciak and Prenowitz [18]. The topic is covered in considerable detail in [19]. Jantosciak has noted that join spaces are a special case of hypergroups =-=[9]-=-. In recent years there has been considerable activity on fuzzy algebraic hyperstructures. Corsini and Zahedi introduced some crisp algebraic hyperstructures related to fuzzy sets [5, 6, 30]. More rec... |

2 |
On reversible Hv groups
- Spartalis
- 1994
(Show Context)
Citation Context ...ave (a Op b) Opc = a Op (b Op Cp (p-associativity). (ii) For all asX we have aSX = X (R3 reproducibility). We now introduce an additional definition which makes use of Hv associativity (see Spartalis =-=[20, 21]-=- and Vougiouklis [22, 25, 26]). Definition 4.15 Given a hyperoperations: X x X - F(X), the hyperstructure (X, ) is called Lfuzzy$-Hv-hypergroup if it satisfies the following conditions. (i) For all a,... |

2 |
On Hv-rings and Hv-representations
- Vougiouklis
- 1999
(Show Context)
Citation Context ...b Op Cp (p-associativity). (ii) For all asX we have aSX = X (R3 reproducibility). We now introduce an additional definition which makes use of Hv associativity (see Spartalis [20, 21] and Vougiouklis =-=[22, 25, 26]-=-). Definition 4.15 Given a hyperoperations: X x X - F(X), the hyperstructure (X, ) is called Lfuzzy$-Hv-hypergroup if it satisfies the following conditions. (i) For all a, b, c,psX we have (asb)sc ..p... |

2 |
Fuzzy ideals on a distributive lattice
- Yuan, Wu
- 1990
(Show Context)
Citation Context ...ntroduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings are studied in [7]. Fuzzy lattices have been introduced in =-=[27]-=- and studied in more detail in [1]. L-fuzzy lattices have been studied in [23] and [24]. Fuzzy intervals (in a lattice theoretic sense, as a special case of fuzzy lattices) are studied in [13]. In thi... |

2 |
On the prime, primary and maximal sybhypermodules
- Zahedi, Ameri
- 1998
(Show Context)
Citation Context ...e of hypergroups [9]. In recent years there has been considerable activity on fuzzy algebraic hyperstructures. Corsini and Zahedi introduced some crisp algebraic hyperstructures related to fuzzy sets =-=[5, 6, 30]-=-. More recently, the concept of fuzzy hypergroups has been introduced in [4]; in fact the closely related concept of fuzzy polygroups has been introduced somewhat earlier in [28, 29]. Fuzzy hyperrings... |

2 | Enlarging Hv-structures”. Algebras and combinatorics (Hong Kong - Vougiouklis - 1997 |

1 | Convexity in Lattices and an Isotone Hyperoperation
- Kehagias, Konstantinidou
- 2001
(Show Context)
Citation Context ...is research to: (a) the study of (X,,p) for a fixed p, (b) the study of the family {(X, *p))pcX, (c) the study of analogous order and distributivity properties of the L-fuzzy join space (X, ,). 2. In =-=[12]-=- we have examined convexity properties of crisp join hyperoperations. One may, analogously, explore fuzzy convexity properties of the L-fuzzyshyperoperation. 3. Choosing a particular pair a, bsX sever... |

1 |
Sur le role de la notion d’hypergroups dans l’ etude des groupes non-abeliens
- Marty
- 1935
(Show Context)
Citation Context ...c structure which is related to several themes from the theories of hypergroups, join spaces and fuzzy sets. Let us first briefly review some work on these themes. Hypergroups have been introduced in =-=[14]-=- and studied extensively by many mathematicians. A comprehensive review of the theory of hypergroups appears in [3]. Join spaces have been introduced by Jantosciak and Prenowitz [18]. The topic is cov... |

1 |
Enlarging Hv-structures". Algebras and combinatorics (Hong Kong
- Vougiouklis
- 1997
(Show Context)
Citation Context ...b Op Cp (p-associativity). (ii) For all asX we have aSX = X (R3 reproducibility). We now introduce an additional definition which makes use of Hv associativity (see Spartalis [20, 21] and Vougiouklis =-=[22, 25, 26]-=-). Definition 4.15 Given a hyperoperations: X x X - F(X), the hyperstructure (X, ) is called Lfuzzy$-Hv-hypergroup if it satisfies the following conditions. (i) For all a, b, c,psX we have (asb)sc ..p... |

1 | The Lattice of Fuzzy Intervals and Su#cient Conditions for Its Distributivity - Kehagias |

1 | On reversible H v groups - Spartalis - 1994 |

1 | Enlarging H v -structures". Algebras and combinatorics (Hong Kong - Vougiouklis - 1997 |