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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.

06D20 Heyting algebras (lattice-theoretic aspects)
06F15 Ordered groups
Full Text: DOI
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