special elements in a lattice
Let L be a lattice and a∈L is said to be
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distributive if a∨(b∧c)=(a∨b)∧(a∨c),
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standard if b∧(a∨c)=(b∧a)∨(b∧c), or
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neutral if (a∧b)∨(b∧c)∨(c∧a)=(a∨b)∧(b∨c)∧(c∨a)
for all b,c∈L. There are also dual notions of the three types mentioned above, simply by exchanging ∨ and ∧ in the definitions. So a dually distributive element a∈L is one where a∧(b∨c)=(a∧b)∨(a∧c) for all b,c∈L, and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.
Remarks For any a∈L, suppose P is the property in L such that a∈P iff a∨b=a∨c and a∧b=a∧c imply b=c for all b,c∈L.
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A standard element is distributive. Conversely, a distributive satisfying P is standard.
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A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that satisfies P is neutral.
References
- 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
- 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
Title | special elements in a lattice |
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Canonical name | SpecialElementsInALattice |
Date of creation | 2013-03-22 16:42:29 |
Last modified on | 2013-03-22 16:42:29 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06B99 |
Defines | distributive element |
Defines | standard element |
Defines | neutral element |
Defines | dually distributive |
Defines | dually standard |