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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
##### Keywords:
semilattice; antitone involution
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