special elements in a lattice
Let be a lattice and is said to be
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distributive if ,
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standard if , or
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neutral if
for all . There are also dual notions of the three types mentioned above, simply by exchanging and in the definitions. So a dually distributive element is one where for all , and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.
Remarks For any , suppose is the property in such that iff and imply for all .
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A standard element is distributive. Conversely, a distributive satisfying is standard.
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A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that satisfies is neutral.
References
- 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
- 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).