Difference between revisions of "Dimension function"
From Encyclopedia of Mathematics
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| − | An integer-valued function | + | 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]]]). | + | 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]]]). |
| + | |||
| + | See also: [[Rank of a partially ordered set]]; [[Dimension 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> | + | <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> | ||
| + | |||
| + | {{TEX|done}} | ||
Latest revision as of 20:59, 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; Dimension 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
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