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Lattice-inadmissible incidence structures

Frantisek Machala, Vladimír Slezák (2004)

Discussiones Mathematicae - General Algebra and Applications

Join-independent and meet-independent sets in complete lattices were defined in [6]. According to [6], to each complete lattice (L,≤) and a cardinal number p one can assign (in a unique way) an incidence structure J L p of independent sets of (L,≤). In this paper some lattice-inadmissible incidence structures are founded, i.e. such incidence structures that are not isomorphic to any incidence structure J L p .

Lattices and semilattices having an antitone involution in every upper interval

Ivan Chajda (2003)

Commentationes Mathematicae Universitatis Carolinae

We study -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.

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