Weakly regular lattices
Ivan Chajda · Czech digital mathematics library · 1985
In [1] G. Birkhoff proposed the following problem: to find necessary and sufficient conditions under which there exists a one-to-one correspondence between congruences and ideals of a lattice.More precisely, we have three problems: under which conditions, on a lattice L, each ideal of L is a kernel of (a) at least one congruence on L;(b) at most one congruence on L;(c) just one congruence on L. Problems (a) and (c) were solved by J. Hashimoto [6].It is well known that (a) is equivalent to the distributivity of L and (c) is equivalent to the following condition:(c*) L is a distributive, relatively complemented lattice with zero element.As far as (b) is concerned, it is not too complicated to prove that each lattice L satisfying (b) has a zero (i.e. a least) element 0, see [5].Use the terminology of [4]:An algebra 91 with a constant 0 is weakly regular if each two congruence