Dimension function

An integer-valued function on a lattice (that is, a mapping ) that satisfies the conditions: 1) for any ; and 2) if is an elementary interval in , then . 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]).

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)