special elements in a lattice

Let $L$ be a lattice and $a\in L$ is said to be

• distributive if $a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)$,

• standard if $b\wedge(a\vee c)=(b\wedge a)\vee(b\wedge c)$, or

• neutral if $(a\wedge b)\vee(b\wedge c)\vee(c\wedge a)=(a\vee b)\wedge(b\vee c)\wedge(c\vee a)$

for all $b,c\in L$. There are also dual notions of the three types mentioned above, simply by exchanging $\vee$ and $\wedge$ in the definitions. So a dually distributive element $a\in L$ is one where $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ for all $b,c\in L$, and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.

Remarks For any $a\in L$, suppose $P$ is the property in $L$ such that $a\in P$ iff $a\vee b=a\vee c$ and $a\wedge b=a\wedge c$ imply $b=c$ for all $b,c\in L$.

• A standard element is distributive. Conversely, a distributive satisfying $P$ is standard.

• 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 SpecialElementsInALattice 2013-03-22 16:42:29 2013-03-22 16:42:29 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 06B99 distributive element standard element neutral element dually distributive dually standard