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join irreducibility
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(Definition)
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An element  in a lattice  is said to be join irreducible iff  is not a bottom element, and, whenever  , then  or  . Dually,  is meet irreducible iff  is not a top element, and, whenever
 , then  or  . If  is both join and meet irreducible, then  is said to be irreducible. Any atom in a lattice is join irreducible.
Example. In the lattice diagram (Hasse diagram) below,
 are join irreducible, while  are meet irreducible. Since  is both join and meet irreducible, it is irreducible.
From this, we make the observations that in any chain, all the elements except the bottom one are join irreducible. Dually, all the elements except the top one are meet irreducible. An element is join irreducible iff it covers at most one other element. An element is meet irreducible iff it is covered by at most one other element.
Remark. If a lattice satisfies the descending chain condition, then every element can be expressed as a join of join irreducible elements. This statement can be dualized: if a lattice satisfies the ascending chain condition, then every element is the meet of meet irreducible elements.
- 1
- B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
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"join irreducibility" is owned by CWoo.
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(view preamble | get metadata)
| Other names: |
join-irreducible, meet-irreducible |
| Also defines: |
join irreducible, meet irreducible, irreducible |
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Cross-references: meet, ascending chain condition, descending chain condition, chain, Hasse diagram, diagram, atom, join, top, bottom, iff, lattice
There are 7 references to this entry.
This is version 6 of join irreducibility, born on 2006-03-20, modified 2007-09-25.
Object id is 7752, canonical name is JoinIrreducibility.
Accessed 4142 times total.
Classification:
| AMS MSC: | 06B99 (Order, lattices, ordered algebraic structures :: Lattices :: Miscellaneous) |
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Pending Errata and Addenda
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