special elements in a lattice (original) (raw)

Let L be a lattice and a∈L is said to be

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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).