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Lattices and semilattices having an antitone involution in every upper interval. (English) Zbl 1101.06003
Summary: We study \(\vee \)-semilattices and lattices with the greatest element 1 where every interval \([p,1]\) is a lattice with an antitone involution. We characterize these semilattices by means of an induced binary operation, the so-called sectionally antitone involution. This characterization is done by means of identities, thus the classes of these semilattices or lattices form varieties. The congruence properties of these varieties are investigated.

MSC:
06A12 Semilattices
06B05 Structure theory of lattices
06B20 Varieties of lattices
08B10 Congruence modularity, congruence distributivity
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