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complete distributivity (Definition)

A lattice $ L$ is said to be completely distributive if it is a complete lattice such that, for any subset $ \lbrace x_{ij}\rbrace$ of $ L$ indexed by some subset $ K$ of $ I\times J$ where $ (i,j)\in K$ such that $ p_1(K)=I$, where $ p_1:I\times J\to I$ is the projection map onto $ I$, we have

$\displaystyle \bigwedge_{i\in I}(\bigvee_{j\in K(i)} x_{ij})=\bigvee_{f\in A}(\bigwedge_{i\in I} x_{if(i)}),$ (1)

where $ K(i):=\lbrace j\in J\mid (i,j)\in K\rbrace$, and $ A=\lbrace f:I\to J\mid f(i)\in K(i)$ for all $ i\in I\rbrace$.

Remarks.

  1. Clearly, a completely distributive lattice is a distributive lattice (by setting $ I=\lbrace 1\rbrace$ and $ J=\lbrace 1,2\rbrace$ and vice versa).
  2. When one of the index sets $ I,J$ is a singleton, we have the concepts of join infinite distributivity and the meet infinite distributivity, intermediate between those of distributivity and complete distributivity. A lattice $ L$ is join infinite distributive if for any $ x\in L$ and any subset $ \lbrace y_i\rbrace$ of $ L$ indexed by a set $ I$ ($ i\in I$), we have

    $\displaystyle x\land (\bigvee_{i\in I} y_i)=\bigvee_{i\in I} (x\land y_i).$

    That is, the join operation is distributive over the meet of a set, possibly infinite, of elements in $ L$. Dually, a lattice is meet infinite distributive if

    $\displaystyle x\lor (\bigwedge_{i\in I} y_i)=\bigwedge_{i\in I} (x\lor y_i).$
  3. One more intermediate concept is to take both $ I$ and $ J$ to be countable, and we get countable distributivity. In this case, one of the index sets, say $ I$, can be rearranged so that for each $ j\in J$, the set $ I(j)=\lbrace 1,2,\ldots,n(j)\rbrace$. A lattice that satisfies both countable distributivity laws must be a countably complete lattice.
  4. A weaker form of 2. above is the concept of meet continuity and its dual, join continuity. Again, one of $ I,J$ is a singleton, say $ J=\lbrace x\rbrace$. But we are requiring that $ I$ be a directed set, or dually a filtered set. Specifically, if

    $\displaystyle x\land (\bigvee_{i\in I} y_i)=\bigvee_{i\in I} (x\land y_i),$

    where $ \lbrace y_i\mid i\in I\rbrace$ is a directed set, then it is said to be meet continuous. Dually, a join continuous lattice $ L$ is one where the following distributivity

    $\displaystyle x\lor (\bigwedge_{i\in I} y_i)=\bigwedge_{i\in I} (x\lor y_i)$

    is satisfied for every element $ x$, and every filtered set $ \lbrace y_i\mid i\in I\rbrace$ in $ L$.

  5. A lattice that satisfies any of meet infinite distributivity, join infinite distributivity, meet continuity, or join continuity individually must be complete inherently.
  6. In some literature, completeness assumption is not required, so that the equation (1) above is conditionally defined. In other words, the equation is defined only when each of the arbitrary join and meet operations is defined in the first place.
  7. Another generalization is the so-called $ (\mathfrak{m},\mathfrak{n})$-distributivity, where $ \mathfrak{m}$ and $ \mathfrak{n}$ are cardinal numbers. Specifically, a lattice $ L$ is $ (\mathfrak{m},\mathfrak{n})$-distributive if it is complete and equation (1) is true whenever $ I$ has cardinality $ \le \mathfrak{m}$ and each $ K(i)$ has cardinality $ \le \mathfrak{n}$ for each $ i\in I$.

Bibliography

1
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).



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Other names:  join infinite distributivity, meet infinite distributivity
Also defines:  completely distributive, join infinitely distributive, meet infinitely distributive, countably distributive, $(m,n)$-distributive
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Cross-references: cardinality, cardinal numbers, place, arbitrary join, equation, complete, join continuous, meet continuous, filtered set, directed set, countably complete lattice, countable, meet, operation, distributive, infinite, join, distributivity, singleton, index sets, distributive lattice, onto, projection map, indexed by, subset, complete lattice, lattice
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This is version 11 of complete distributivity, born on 2006-02-18, modified 2008-02-27.
Object id is 7636, canonical name is CompleteDistributivity.
Accessed 3772 times total.

Classification:
AMS MSC: 06D10 (Order, lattices, ordered algebraic structures :: Distributive lattices :: Complete distributivity)
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