Abstract. Lattices $L$ with $0$ are investigated such that each ideal of $L$ is of the form $\{x\colon \langle x,0\rangle\in \tau\} $ for some tolerance relation $\tau $. We show that $L$ has this property iff for any $b\in L$ and every unary lattice polynomial $p(x)$ with $p(0)=0$ we have $p(b)\le b$. If, in addition, $L$ is atomic then the ideal generated by any finite set of atoms in $L$ is shown to be a Boolean sublattice of $L$.

AMS Subject Classification (1991): 06B05, 06B10, 06B15

Keyword(s): Lattice, 0, tolerance, kernel, ideal, distributivity

Received March 16, 1995. (Registered under 5668/2009.)