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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:
 [1] Anderson, M. andFeil, T.,Latice-Ordered Groups. Reidel, Dordrecht-Boston-Lancaster-Tokyo, 1988. [2] BalachandrAN, V. K.,A characterization of ??-rings of subsets. Fund. Math.41 (1954), 38-41. · Zbl 0055.28001 [3] Balbes, R. andDwinger, P.,Distributive Lattices. University of Missouri Press, Columbia, 1974. [4] Bigard, A., Keimel, K. andWolfenstein, S.,Groupes et Anneaux Reticules (Lecture Notes in Mathematics608), Springer-Verlag, Berlin-Heidelberg-New York, 1977. [5] B?chi, J. R.,Representation of complete lattices by sets. Portug. Math.2 (1952), 149-186. [6] Conrad, P. F.,The lattice of all convex ?-subgroups of a lattice-ordered group. Czech. Math. J.15 (1965), 101-123. · Zbl 0135.06301 [7] Cornish, W. H.,Normal lattices. J. Australian Math. Soc.14 (1972), 200-215. · Zbl 0247.06009 · doi:10.1017/S1446788700010041 [8] Crawley, P. andDilworth, R. P.,Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, 1973. [9] Gratzer, G.,General Lattice Theory. Academic Press, New York, 1978. [10] Martinez, J.,Unique factorization in partially ordered sets. Proc. Amer. Math. Soc.33 (1972), 213-220. · Zbl 0241.06007 · doi:10.1090/S0002-9939-1972-0292723-5 [11] McKenzie, R. N., McNulty, G. F. andTaylor, W. F.,Algebras, Lattices, Varieties (Volume I). Wadsworth and Brooks/Cole, Monterey, 1987. · Zbl 0611.08001 [12] Mandelker, M.,Relative annihilators in lattices. Duke Math. J.37 (1970), 377-386. · Zbl 0206.29701 · doi:10.1215/S0012-7094-70-03748-8 [13] Monteiro, A.,V Arithm?tique des filtres et les espaces topologiques. De Segundo Symposium de Mathematicas-Villavicencio, Mendoza; Buenos Aires, 1954, 129-162. [14] Monteiro, A.,L’Arithm?tique des filtres et les espaces topologiques I?II. Notas de L?gica Matem?tica, Vol. 29-30, 1974. [15] Raney, G. N.,Completely distributive complete lattices. Proc. Amer. Math. Soc.3 (1952), 677-680. · Zbl 0049.30304 · doi:10.1090/S0002-9939-1952-0052392-3 [16] Snodgrass, J. T. andTsinakis, C.,The finite basis theorem for relatively normal lattices. Preprint. [17] Steinberg, S. A.,Finitely-valued f-modules. Pacific J. Math.40 (1972), 723-737. · Zbl 0218.16008 [18] Zaanen, A. C.,Riesz Spaces II. North-Holland, Amsterdam-New York-Oxford, 1983. · Zbl 0519.46001
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