# Dimension 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 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=40061