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On the special context of independent sets

Vladimír Slezák — 2001

Discussiones Mathematicae - General Algebra and Applications

In this paper the context of independent sets J L p is assigned to the complete lattice (P(M),⊆) of all subsets of a non-empty set M. Some properties of this context, especially the irreducibility and the span, are investigated.

Lattice-inadmissible incidence structures

Frantisek MachalaVladimír 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 .

Negation in bounded commutative D R -monoids

Jiří RachůnekVladimír Slezák — 2006

Czechoslovak Mathematical Journal

The class of commutative dually residuated lattice ordered monoids ( D R -monoids) contains among others Abelian lattice ordered groups, algebras of Hájek’s Basic fuzzy logic and Brouwerian algebras. In the paper, a unary operation of negation in bounded D R -monoids is introduced, its properties are studied and the sets of regular and dense elements of D R -monoids are described.

Join-closed and meet-closed subsets in complete lattices

František MachalaVladimír Slezák — 2004

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

To every subset A of a complete lattice L we assign subsets J ( A ) , M ( A ) and define join-closed and meet-closed sets in L . Some properties of such sets are proved. Join- and meet-closed sets in power-set lattices are characterized. The connections about join-independent (meet-independent) and join-closed (meet-closed) subsets are also presented in this paper.

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