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Finite orders and their minimal strict completion lattices

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Bordalo G., Monjardet B.

Discussiones Mathematicae - General Algebra and Applications

2003

tom 23

nr 2

85-100

Whereas the Dedekind-MacNeille completion D(P) of a poset P is the minimal lattice L such that every element of L is a join of elements of P, the minimal strict completion D(P)∗ is the minimal lattice L such that the poset of join-irreducible elements of L is isomorphic to P. (These two completions are the same if every element of P is join-irreducible). In this paper we study lattices which are minimal strict completions of finite orders. Such lattices are in one-to-one correspondence with finite posets. Among other results we show that, for every finite poset P, D(P)∗ is always generated by its doubly-irreducible elements. Furthermore, we characterize the posets P for which D(P)∗ is a lower semimodular lattice and, equivalently, a modular lattice.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Bordalo G.

1 Monjardet B.

1 2003

Wyniki wyszukiwania

Finite orders and their minimal strict completion lattices