### 2-normalization of lattices

Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v\left(t\right)\ge 0$. For $k\ge 1$, an identity $s\approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s=t$ or both $s$ and $t$ have value $\ge k$. Taking $k=1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least...