Completions of Pseudo Ordered Sets
A pseudo ordered set ( X ,≤) is a set X with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set X , a partially ordered set Γ( X ) called the covering poset. Taking any completion ( C , f ) of the covering poset Γ( X ), and a special equivalence relation 𝜃...
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| Published in | Order (Dordrecht) Vol. 39; no. 1; pp. 95 - 111 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | A pseudo ordered set (
X
,≤) is a set
X
with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set
X
, a partially ordered set Γ(
X
) called the covering poset. Taking any completion (
C
,
f
) of the covering poset Γ(
X
), and a special equivalence relation
𝜃
on this completion, yields a completion
C
/
𝜃
of the pseudo ordered set
X
. The case when (
C
,
f
) is the MacNeille completion of Γ(
X
) gives the pseudo MacNeille completion of
X
. |
|---|---|
| AbstractList | A pseudo ordered set (
X
,≤) is a set
X
with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set
X
, a partially ordered set Γ(
X
) called the covering poset. Taking any completion (
C
,
f
) of the covering poset Γ(
X
), and a special equivalence relation
𝜃
on this completion, yields a completion
C
/
𝜃
of the pseudo ordered set
X
. The case when (
C
,
f
) is the MacNeille completion of Γ(
X
) gives the pseudo MacNeille completion of
X
. A pseudo ordered set (X,≤) is a set X with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set X, a partially ordered set Γ(X) called the covering poset. Taking any completion (C,f) of the covering poset Γ(X), and a special equivalence relation ðoef on this completion, yields a completion C/ðoef of the pseudo ordered set X. The case when (C,f) is the MacNeille completion of Γ(X) gives the pseudo MacNeille completion of X. |
| Author | Cruz-Quinones, Maria D. Harding, John |
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| Cites_doi | 10.1007/s000120050189 10.1007/BF02945138 10.1007/978-3-0348-7633-9 10.1007/BF01900297 10.1023/B:CMAJ.0000027267.89851.85 10.1007/BF02944982 10.1090/memo/0121 10.1007/978-3-540-77664-2_18 |
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| Keywords | Pseudo order Completion Trellis MacNeille completion |
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| References | Gladstien (CR6) 1973; 3 Skala (CR11) 1971; 1 CR2 Schmidt (CR10) 1956; 7 Grätzer (CR5) 1978 Fried, Grätzer (CR4) 1976; 2 CR7 Banaschewski (CR1) 1956; 2 Fried (CR3) 1970; 13 Parameshwara Bhatta, Shashirekha (CR8) 2000; 44 Parameshwara Bhatta, Shashirekha (CR9) 2004; 54 CR12 E Fried (9565_CR4) 1976; 2 9565_CR12 B Banaschewski (9565_CR1) 1956; 2 G Grätzer (9565_CR5) 1978 9565_CR2 E Fried (9565_CR3) 1970; 13 J Schmidt (9565_CR10) 1956; 7 S Parameshwara Bhatta (9565_CR9) 2004; 54 9565_CR7 K Gladstien (9565_CR6) 1973; 3 S Parameshwara Bhatta (9565_CR8) 2000; 44 HL Skala (9565_CR11) 1971; 1 |
| References_xml | – volume: 44 start-page: 305 year: 2000 end-page: 308 ident: CR8 article-title: A characterization of completeness for trellises publication-title: Algebra Univers. doi: 10.1007/s000120050189 contributor: fullname: Shashirekha – volume: 2 start-page: 501 issue: 4 year: 1976 end-page: 512 ident: CR4 article-title: Partial and free weakly associative lattices publication-title: Houston J. Math. contributor: fullname: Grätzer – volume: 3 start-page: 341 year: 1973 end-page: 344 ident: CR6 article-title: A characterization of complete trellises of finite length publication-title: Algebra Universalis doi: 10.1007/BF02945138 contributor: fullname: Gladstien – year: 1978 ident: CR5 publication-title: General Lattice Theory doi: 10.1007/978-3-0348-7633-9 contributor: fullname: Grätzer – volume: 2 start-page: 35 year: 1956 end-page: 46 ident: CR1 article-title: Hüllensysteme und Erweiterungen von Quasi-Ordnungen publication-title: Z. Math. Logik Grund. Math. contributor: fullname: Banaschewski – volume: 13 start-page: 151 year: 1970 end-page: 164 ident: CR3 article-title: Tournaments and non-associative lattices publication-title: Ann. Univ. Sci. Budapest Eotvos Sect. Math. contributor: fullname: Fried – ident: CR7 – volume: 7 start-page: 241 year: 1956 end-page: 249 ident: CR10 article-title: Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge publication-title: Archiv. Math. doi: 10.1007/BF01900297 contributor: fullname: Schmidt – ident: CR2 – volume: 54 start-page: 267 issue: 1 year: 2004 end-page: 272 ident: CR9 article-title: Some characterizations of completeness for trellises in terms of joins of cycles publication-title: Czech. Math. J. doi: 10.1023/B:CMAJ.0000027267.89851.85 contributor: fullname: Shashirekha – ident: CR12 – volume: 1 start-page: 218 year: 1971 end-page: 233 ident: CR11 article-title: Trellis theory publication-title: Algebra Univers. doi: 10.1007/BF02944982 contributor: fullname: Skala – volume: 2 start-page: 35 year: 1956 ident: 9565_CR1 publication-title: Z. Math. Logik Grund. Math. contributor: fullname: B Banaschewski – volume: 13 start-page: 151 year: 1970 ident: 9565_CR3 publication-title: Ann. Univ. Sci. Budapest Eotvos Sect. Math. contributor: fullname: E Fried – volume-title: General Lattice Theory year: 1978 ident: 9565_CR5 doi: 10.1007/978-3-0348-7633-9 contributor: fullname: G Grätzer – ident: 9565_CR12 doi: 10.1090/memo/0121 – ident: 9565_CR2 – volume: 2 start-page: 501 issue: 4 year: 1976 ident: 9565_CR4 publication-title: Houston J. Math. contributor: fullname: E Fried – volume: 1 start-page: 218 year: 1971 ident: 9565_CR11 publication-title: Algebra Univers. doi: 10.1007/BF02944982 contributor: fullname: HL Skala – volume: 54 start-page: 267 issue: 1 year: 2004 ident: 9565_CR9 publication-title: Czech. Math. J. doi: 10.1023/B:CMAJ.0000027267.89851.85 contributor: fullname: S Parameshwara Bhatta – volume: 7 start-page: 241 year: 1956 ident: 9565_CR10 publication-title: Archiv. Math. doi: 10.1007/BF01900297 contributor: fullname: J Schmidt – volume: 3 start-page: 341 year: 1973 ident: 9565_CR6 publication-title: Algebra Universalis doi: 10.1007/BF02945138 contributor: fullname: K Gladstien – volume: 44 start-page: 305 year: 2000 ident: 9565_CR8 publication-title: Algebra Univers. doi: 10.1007/s000120050189 contributor: fullname: S Parameshwara Bhatta – ident: 9565_CR7 doi: 10.1007/978-3-540-77664-2_18 |
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| Snippet | A pseudo ordered set (
X
,≤) is a set
X
with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set
X
, a partially... A pseudo ordered set (X,≤) is a set X with a binary relation ≤ that is reflexive and antisymmetric. We associate to a pseudo ordered set X, a partially ordered... |
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| SubjectTerms | Algebra Discrete Mathematics Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures Pallets Set theory |
| Title | Completions of Pseudo Ordered Sets |
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