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Difference between revisions of "Dimension function"

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An integer-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324701.png" /> on a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324702.png" /> (that is, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324703.png" />) that satisfies the conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324704.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324705.png" />; and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324706.png" /> is an [[Elementary interval|elementary interval]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324707.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032470/d0324708.png" />. For a lattice in which all bounded chains are finite, the existence of a dimension function is equivalent to the modular property.
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An integer-valued function $d$ on a [[lattice]] $L$ (that is, a mapping $D : L \rightarrow \mathbf{Z}$) that satisfies the conditions: 1) $d(x \vee y) + d(x \wedge y) = d(x) + d(y)$ for any $x,y \in L$; and 2) if $[x,y]$ is an [[elementary interval]] in $L$, then $d(y) = d(x)+1$. For a lattice in which all bounded chains are finite, the existence of a dimension function is equivalent to the [[Modular lattice|modular property]].
  
There is also a more general definition of a dimension function on an orthomodular lattice or on an orthomodular partially ordered set, where the values of the dimension function can be arbitrary real numbers, or even functions (see [[#References|[3]]]).
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There is also a more general definition of a dimension function on an [[orthomodular lattice]] or on an orthomodular partially ordered set, where the values of the dimension function can be arbitrary real numbers, or even functions (see [[#References|[3]]]).
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See also: [[Rank of a partially ordered set]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Kalmbach,  "Orthomodular lattices" , Acad. Press  (1983)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  G. Kalmbach,  "Orthomodular lattices" , Acad. Press  (1983)</TD></TR>
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</table>
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Revision as of 16:22, 18 December 2016

An integer-valued function $d$ on a lattice $L$ (that is, a mapping $D : L \rightarrow \mathbf{Z}$) that satisfies the conditions: 1) $d(x \vee y) + d(x \wedge y) = d(x) + d(y)$ for any $x,y \in L$; and 2) if $[x,y]$ is an elementary interval in $L$, then $d(y) = d(x)+1$. For a lattice in which all bounded chains are finite, the existence of a dimension function is equivalent to the modular property.

There is also a more general definition of a dimension function on an orthomodular lattice or on an orthomodular partially ordered set, where the values of the dimension function can be arbitrary real numbers, or even functions (see [3]).

See also: Rank of a partially ordered set.

References

[1] L.A. Skornyakov, "Elements of lattice theory" , Hindushtan Publ. Comp. (1977) (Translated from Russian)
[2] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[3] G. Kalmbach, "Orthomodular lattices" , Acad. Press (1983)
How to Cite This Entry:
Dimension function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dimension_function&oldid=13929
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article