Dimension function

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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 property.