Nodal filters in hoop algebras
Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287, 1969 , Fundam Math 69:1–14, 1970 ). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that t...
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| Published in | Soft computing (Berlin, Germany) Vol. 22; no. 21; pp. 7119 - 7128 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
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Springer Berlin Heidelberg
01.11.2018
Springer Nature B.V |
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| Abstract | Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287,
1969
, Fundam Math 69:1–14,
1970
). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that the sets of all nodes are a bounded distributive lattice. Then by define some operations on
NF
(
A
)
, the set of all nodal filters in hoop
A
, we show that
NF
(
A
)
is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. Finally, we investigate the relation among nodal filters and (positive) implicative, obstinate, prime and maximal filters in any hoops. |
|---|---|
| AbstractList | Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287,
1969
, Fundam Math 69:1–14,
1970
). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that the sets of all nodes are a bounded distributive lattice. Then by define some operations on
NF
(
A
)
, the set of all nodal filters in hoop
A
, we show that
NF
(
A
)
is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. Finally, we investigate the relation among nodal filters and (positive) implicative, obstinate, prime and maximal filters in any hoops. Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287, 1969, Fundam Math 69:1–14, 1970). In this paper, we introduce the notions of node and nodal filter in hoops and study some properties of them. First, we prove that the sets of all nodes are a bounded distributive lattice. Then by define some operations on NF(A), the set of all nodal filters in hoop A, we show that NF(A) is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. Finally, we investigate the relation among nodal filters and (positive) implicative, obstinate, prime and maximal filters in any hoops. |
| Author | Borzooei, R. A. Namdar, A. |
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| Cites_doi | 10.1007/s11225-007-9078-1 10.5373/jarpm.1895.120113 10.1017/S0022481200029716 10.1090/S0002-9947-1970-0256952-1 10.3233/IFS-151804 10.4064/fm-69-1-1-14 10.1007/978-94-011-5300-3 10.3233/IFS-151589 10.4064/fm-64-3-257-287 10.1007/s000120050156 10.3233/JIFS-152553 10.3233/IFS-141399 10.1109/ISMVL.2011.9 |
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| Keywords | Heyting algebra Node Hertz algebra Hilbert algebra BCK-algebra Kleene algebra Semi-De Morgan algebra Hoop Nodal filter |
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| References | Borzooei, Aaly Kologani (CR7) 2015; 29 Meng, Jun (CR15) 1994 Aglianó, Ferreirim, Montagna (CR1) 2007; 87 Balbes, Horn (CR2) 1970; 148 Bakhshi (CR3) 2016; 30 Boicescu, Filipoiu, Georgescu, Rudeanu (CR4) 1991 Diego (CR11) 1966; 21 CR14 Bosbach (CR8) 1969; 64 Georgescu, Leustean, Preoteasa (CR12) 2005; 11 CR10 CR21 CR20 Borzooei, Aaly Kologani (CR6) 2014; 6 Bosbach (CR9) 1970; 69 Hájek (CR13) 1998 Blok, Ferreirim (CR5) 2000; 43 Piciu (CR17) 2007 Sankappanavar (CR19) 1987; 52 Varlet (CR22) 1973; 14 Porta (CR18) 1981; 40 Namdar, Borzooei, Borumand Saeid, Aaly Kologani (CR16) 2017; 32 H Porta (2986_CR18) 1981; 40 G Georgescu (2986_CR12) 2005; 11 V Boicescu (2986_CR4) 1991 HP Sankappanavar (2986_CR19) 1987; 52 RA Borzooei (2986_CR6) 2014; 6 B Bosbach (2986_CR8) 1969; 64 2986_CR14 2986_CR16 P Aglianó (2986_CR1) 2007; 87 2986_CR10 J Meng (2986_CR15) 1994 2986_CR21 M Bakhshi (2986_CR3) 2016; 30 A Diego (2986_CR11) 1966; 21 D Piciu (2986_CR17) 2007 WJ Blok (2986_CR5) 2000; 43 RA Borzooei (2986_CR7) 2015; 29 P Hájek (2986_CR13) 1998 2986_CR20 JC Varlet (2986_CR22) 1973; 14 R Balbes (2986_CR2) 1970; 148 B Bosbach (2986_CR9) 1970; 69 |
| References_xml | – year: 1991 ident: CR4 publication-title: Łukasiewicz–Moisil algebras. 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| Snippet | Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287,
1969
, Fundam Math... Hoop algebras or hoops are naturally ordered commutative residuated integral monoids, introduced by Bosbach (Fundam Math 64:257–287, 1969, Fundam Math 69:1–14,... |
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| Title | Nodal filters in hoop algebras |
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