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Finite-valued algebraic lattices. (English) Zbl 0806.06011
Let $$L$$ be a distributive algebraic lattice. Then $$L$$ is said to be finite-valued if for every compact element $$c\in L$$, the poset of meet- irreducible elements not exceeding $$c$$ has finitely many maximal elements. It is a classical result that a bialgebraic (= algebraic and dually algebraic) distributive lattice is finite-valued. The converse does not hold in general [see J. Martinez, Proc. Am. Math. Soc. 33, 213-220 (1972; Zbl 0241.06007)].
The principal result of the paper under review says that the converse of the above classical result is possible under some additional assumptions. More precisely, let $$L$$ be an algebraic distributive lattice whose compact elements form a sublattice of whose meet-irreducible elements form a root-system. (A root-system is a poset for which the principal order-filter $$[p)=\{x\in P$$: $$p\leq x\}$$ is a chain for all $$p\in P$$.) Let $$L$$ be finite-valued. Then $$L$$ is bialgebraic. As a corollary of this result one can derive the following known theorem [see P. Conrad, Czech. Math. J. 15, 101-123 (1965; Zbl 0135.063)]: The congruence lattice $$\text{Con}(L)$$ of an $$l$$-group $$L$$ is finite-valued if and only if each positive element of $$L$$ is a finite product of pairwise orthogonal elements.

MSC:
 06D20 Heyting algebras (lattice-theoretic aspects) 06F15 Ordered groups
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References:
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