On the number of join-irreducibles in a congruence
representation of a finite distributive lattice |
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Authors: | Email author" target="_blank">G?Gr?tzerEmail author F?Wehrung |
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Institution: | (1) Department of Mathematics, University of Manitoba, ., R3T 2N2 Winnipeg, MN, Canada;(2) CNRS ESA 6081, Université de Caen, Campus II, Département de Mathématiques, BP 5186, 14032 Caen Cedex, France |
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Abstract: | For a finite lattice L, let $ \trianglelefteq_L $ denote
the reflexive and transitive closure of the join-dependency relation on L,
defined on the set J(L) of all join-irreducible elements of L.
We characterize the relations of the form $ \trianglelefteq_L $, as follows:
Theorem. Let $ \trianglelefteq $ be a quasi-ordering on
a finite set P. Then the following conditions are
equivalent:(i) There exists a finite lattice L such that $ \langle J(L), \trianglelefteq_L $ is
isomorphic to the quasi-ordered set $ \langle P, \trianglelefteq \rangle $.(ii) $ |\{x\in P|p \trianglelefteq x\}| \neq 2 $, for any $ p \in P $.For a finite lattice L,
let $ \mathrm{je}(L) = |J(L)|-|J(\mathrm{Con} L)| $ where Con L
is the congruence lattice of L. It is well-known that the
inequality $ \mathrm{je}(L) \geq 0 $ holds. For a finite distributive
lattice D, let us define the join-
excess function:$ \mathrm{JE}(D) =\mathrm{min(je} (L) | \mathrm{Con} L \cong D). $We provide a formula for computing the join-excess function of a finite distributive lattice
D. This formula implies that $ \mathrm{JE}(D) \leq (2/3)| \mathrm{J}(D)|$
, for any finite distributive lattice D;
the constant 2/3 is best possible.A special case of this formula gives a characterization of congruence lattices of finite
lower bounded lattices.Dedicated to the memory of Gian-Carlo Rota |
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Keywords: | 06B10 06B15 |
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