You are here
Homespecial elements in a lattice
Primary tabs
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).
Mathematics Subject Classification
06B99 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections