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Distributive lattices whose congruence lattice is Stone. (English) Zbl 0848.06007

Summary: Using Priestley’s topological duality we characterize bounded distributive lattices with \((L_n)\)- and relative \((L_n)\)-congruence lattices. In particular, characterizations of bounded distributive lattices with Stone and relative Stone congruence lattices are obtained. Using these descriptions we derive some results of T. Katriňák [“Notes on Stone lattices. II”, Mat. Čas. Slov. Akad. Vied. 17, 20-37 (1967)], D. Thomas [Problems in functional analysis, Ph. D. thesis (Oxford, 1976)], M. Haviar [Demonstr. Math. 24, No. 1/2, 247-261 (1991; Zbl 0771.06003)], and M. Haviar and T. Katriňák [Acta Sci. Math. 51, 81-91 (1987; Zbl 0646.06005)]. In the last section we discuss questions concerning the relation between completeness of a bounded distributive lattice and its minimal Boolean completion. This is connected with a problem of Thomas [loc. cit.].

MSC:

06D05 Structure and representation theory of distributive lattices
06D15 Pseudocomplemented lattices
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