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 $(a\\wedge b)\\vee(b\\wedge c)\\vee(c\\wedge a)=(a\\vee b)\\wedge(b\\vee c)\\wedge(c\\veea)$

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).

单词	SpecialElementsInALattice
释义	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 $(a\\wedge b)\\vee(b\\wedge c)\\vee(c\\wedge a)=(a\\vee b)\\wedge(b\\vee c)\\wedge(c\\veea)$ 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).
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