Article
Kyungpook Mathematical Journal 2020; 60(1): 1-20
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
On the Representations of Finite Distributive Lattices
Mark Siggers
Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : mhsiggers@knu.ac.kr
Received: January 24, 2017; Revised: January 16, 2020; Accepted: January 16, 2020
Abstract
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
A simple but elegant result of Rival states that every sublattice
Applying this in the case that ℘ is a product of a finite set of chains, we get a one-to-one correspondence
between the sublattices of ℘ and the preorders spanned by a canonical sublattice
of ℘.
We then show that is asymmetric. This yields a one-to-one correspondence between the tight sublattices of ℘ and the posets spanned by its poset
With this we recover and extend, among other classical results, the correspondence derived from results of Birkhoff and Dilworth, between the tight embeddings of a finite distributive lattice
Keywords: finite distributive lattice, representation, embedding, product of chains.
1. Introduction
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
All lattices considered in this paper are finite and distributive. For very basic notation, definitions, and concepts we refer the reader to [4]; many basic definitions are also given at the start of Section 2.
Classical results of Birkhoff and Dilworth, which we review in more detail in Section 2, show that any finite distributive lattice
Our main goal in this paper is to reverse the point of view of this correspondence: instead of cataloguing the various embeddings of a particular lattice
Starting with the product of a set
of chains,
by removing their minimum elements. One main idea is that, for a tight sublattice
But what if we consider non-tight embeddings/sublattices? One can see two examples of non-tight embeddings in the bottom half of Figure 2. As every lattice has a tight embedding into products of chains, non-tight embeddings are seldom considered. However, when we look to catalogue the sublattices of given product of chains, it is natural to consider them. In the paper [9], out of which this paper grew, we found it useful to consider non-tight sublattices of products of chains. We needed a characterisation of
In Section 3, we use a result of Rival [8] which characterises the sublattices of ℘, to construct, for each sublattice of ℘, a that we get from
This preorder is an analogue of . Results similar to those is this section can by found with a different presentation in the recent article [7] of Retakh and Saks.
In Section 5, we return to the point of view of embeddings of a given lattice, and extend Birkhoff’s classical correspondence between the tight embeddings of a lattice
In [5], Koh used a clever construction to show that any distributive lattice of maximal antichains of some poset
for a given
2. Notation and Background
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
A lattice embedding of chains
of disjoint chains, the product
In his famous representation theorem in [1], Birkhoff showed that where
, for a poset
Theorem 2.1.([1])
Observing that a downset in is join-irreducible if and only if it has a unique maximal element, one can easily show that
is an isomorphism. Thus
. This immediatly yields a one-to-one correspondence,
For a chain decomposition of a poset let
be the family of chains we get from the chains in
by adding a new minimum element to each. In [2], Dilworth proved the following embedding theorem.
Theorem 2.2.([2])
.
With Theorem 2.1, this immediately gives the following.
Corollary 2.3
.
In [6], Larson makes explicit a converse to Corollary 2.3, showing essentially the following.
Theorem 2.4.([6])
.
It is a trivial observation that different chain decompositions of
Corollary 2.5
3. The Rival Construction D ℘(L )
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Recalling a result of Rival in the first subsection, we use it in the second subsection to construct the preorder
3.1. Rival’s Theorem
An
3.2. Setup and The Definition of D ℘(L )
For the rest of the paper ℘ is always a product of finite chains
. We will denote the elements of a chain
for
for
Given and
, let
be the subposet of ℘ induced by
Recall that a poset, and in particular , is a reflexive relation. We refer to a reflexive relation simply as a
is any digraph on the elements of
, which contains it. A
The following is our extension of
For any family of chains and any family of irreducible intervals ℐ of
, let
that we get from
by letting (
An (
A subset


For any digraph be the family of downsets of
be the subfamily of non-trivial downsets.
As with posets, it is clear for digraphs that unions and intersections of downsets are downsets, so is indeed a poset for any digraph
is, in fact, the lattice
.
For a poset
Indeed, the isomorphism is the map that drops the appended unit. As implies
With
Indeed ℘ is a lattice, so we may apply Theorem 2.1, and then
Before we prove our main theorem, that for
First though, let’s explain our definition of
3.3. Some Examples
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
A directed graph that extends a poset can be depicted by adding arcs to the Hasse diagram of the poset, though direction must be made explicit on these arcs, as they need not all go up. As the digraphs
The top half of Figure 2 shows the same lattices ℘ = and its spanned extension
The construction of so that
to maintain the isomorphism. Indeed, this ensures that, for example, the set
The third example in the figure shows a non-tight sublattice is not a poset. The fourth example exhibits the necessity of 0 and 1 in
3.4. The Closure of ℐ and Transitivity of D ℘(ℐ)
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
In this subsection we show that
Recall that we included the empty intervals of the form [. By definition then, these are thus contained in any closed family ℐ of irreducible intervals. This is consistant with
. In particular, we assume for any sublattice
Now our technical lemma.
Lemma 3.6
.
[
αi, βj ] ⊆ ∪ℐ.For all x ∈L, xi ≥α ⇒xj > β. There is a ((β + 1)j, αi )-path in D.
The equivalence of (i) and (ii) is immediate as both are equivalent to the statement that there are no elements
On the one hand, consider an ((
By the definition of
On the other hand, assume that there is no such path from (
If
As ℐ
Proposition 3.7
On the one hand, assume that ℐ is closed. Then we can replace (i) of Lemma 3.6 with [
On the other hand, assume that ℐ is not closed. Then there is some [
4. Main Results
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
In this section we give our main results corresponding the sublattices of product of chains to the spanned preorder extensions of
. In the first subsection, we give our main theorem, extending Birkhoff’s representation theorem from
4.1. Main Theorem
Generalising Theorem 2.1 we have the following.
As it simplifies induction, we prove the slightly more general result that , so the isomorphism is given in Lemma 3.5.
In the general case, we first observe that is a subposet of
. Indeed, as
, any downset of
, so
is a subset of
. Since both are ordered by inclusion
.
Now it is enough to show that a bijection. We do this by induction on the size of ℐ. Let [
is a bijection.
We must show for , that
if and only if
To see that is the principle downset
in
, so
4.2. One-to-one Correspondence
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
By Theorem 4.1 we have that any sublattice of ℘ can be expressed as the lattice of non-trivial downsets of some spanned extension
. On the other hand, it is clear that for a spanned extension
the family
is such that is the sublattice ℘ ∪ℐ
Corollary 4.2
.
This solves one of our main goals, ‘reversing the point of view’ of correspondence given in Corollary 2.5.
4.3. Classification of Sublattices
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Recall that a sublattice
It was shown in [6] that every tight sublattice of a product of chains is subdirect. The converse was also claimed, but the proof was flawed: indeed, the lattice
The
Lemma 4.3
L is a {0, 1}-sublattice if and only if 0 is a source (has no in-arcs) in D and 1 is a sink (has no out-arcs). L is subdirect if and only if D has no down edges: those of the form αi → (α − 1)i. L is tight if and only if D is a poset.
We have by Theorem 4.1 that is an isomorphism.
Clearly
T (0 ) = {0 } is a downset ofD if and only if0 is a source inD . Just as clearly,is a downset if and only if
1 is a sink. The result follows.We have
αi → (α − 1)i inD if and only if there is no downsetT (x ) ofD containing (α −1)i but notαi . This is if and only if there is no elementx ∈L such thatxi =α − 1.First assume that
L is tight, and assume, towards contradiction thatD contains a cycle (α 1)i 1 → (α 2)i 2 · · · → (αℓ )i ℓ → (α 1)i 1, for someℓ ≥ 2. We show that no vertexx inL hasxi 1 =α 1, which contradicts the fact thatL is tight. Indeed, ifx did havexi 1 =α 1, thenxi 1 ≥α 1 so by Lemma 3.6xi 2> α 2 soxi 3> α 3 etc., and we get thatxi 1> α 1, a contradiction.
On the other hand, assume that from
Remark 4.4
Though our goal was a correspondence including .
For any preorder , Theorem
is a lattice isomorphism. When
Observing that a spanned extension yields a spanned extension
of , Corollary 4.2 says that the map
5. Back to Embeddings
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
We would now like to extend Corollary 2.5 by fixing a finite distributive lattice such that
. But as neither
are canonical for
rather as a bijective homomorphism
, we get a surjective homomorphism of
by passing to the quotient. As such, every embedding
A for some product
of chains. (Recalling this construction, we get a pointed union of chains
from the parallel sum of the chains of a family
of chains by appending a new zero
of a pointed union
of chains to a preorder
5.1. The Canonical Quotient of D ℘(L )
The
.
As [ to
, and
As , the second fact follows by
Now for any pointed chain decomposition of a preorder
by composing with the quotient map. And while the quotient map
Thus we have the following.
5.2. The Full Correspondence Extending Corollary 2.5
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Fix a lattice , which we view as a pointed chain decomposition
of a pointed union of chains to
Using the isomorphism that we get from Theorem 4.1, we will see that we can write
On the other hand, for any surjective homomorphism from a pointed union of chains, we get, by Fact 5.2, a unique pointed chain decomposition
of a preorder
and Theorem 4.1 gives us an embedding
of , this is an embedding
of
We are ready for our generalisation of Corollary 2.5. Recall that a pointed chain cover of
Theorem 5.3
It is enough to show that the maps , because anything in 〈
and
To see that
Remark 5.4
When removing
6. Lattices of Independent Sets
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Adjusting our construction, we extend and complement results of Koh from [5].
6.1. Background
The correspondence taking a downset of a poset of
of
, the correspondence can be extended to a lattice isomorphism: for antichains
, set
In [5], Koh showed a converse: every finite distributive lattice for some poset
which yields a poset
for every chain decomposition
of the poset
.
There are posets that are not accounted for by Koh’s construction. A small alteration of the reverse point of view of this paper yields a simple generalisation of Koh’s construction. With this we show that every poset
arises from a chain cover of
The alteration of the reverse pont of view yields transitive digraph extensions of a union of tournaments rather than preorder extensions of a pointed union of sets. We start with definitions required to define the antichain lattice of such digraphs.
6.2. Definitions and Setup
A directed path
A subset
This is a much stronger notion of independence than is usually defined for digraphs, it implies independence in the transitive closure of the graph. In fact, no vertex that is in a cycle, including a looped vertex, can be in an independent set.

Let be the family of independent sets of size
, let
For a poset when
Our basic setup now is as follows. Given a product of a family
of
will be the (disjoint)
by removing the loops from all vertices. As we will now be referring to elements of a chain
By Remark 6.5 the are exactly
A digraph if
is a spanning subgraph of
defined in Definition 6.4 has a simpler definition, essentially observed in [5] in the case that
,
The ‘if’ direction is immediate from Definition 6.4 as all arcs of are in
From here, given a digraph of
is a lattice under the above ordering. We will use Dilworth’s chain decomposition in this way in Corollary 6.11, but for now we follow our reverse point of view, and relate such digraphs
6.3. The Alternate Rival Construction A ℘(L )
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Compare the following to Definition 3.2.
Definition 6.7. (A ℘(ℐ) and A ℘(L ))
For a family of chains and any family of irreducible intervals ℐ of
, let
that we get from
by letting
See Figure 4 for examples of
It is easy to see that where
is the first picture in Figure 4. This an example of the base case of Theorem 6.8. To prove Theorem 6.8 we simply observe what we must do to
to maintain this isomorphism when we remove an irreducible interval from ℘. Remove the interval [31, 32] as we do to get the second lattice
. To ensure that, for example,
. This also ensures that such sets as
Theorem 6.8
,
The theorem holds by Lemma 6.6 in the simplest case where . We proceed by induction on the size of
or any spanned extension cannot not create new independent sets, we have that
is a subset of ℘ =
; and so by Lemma 6.6 it is a subposet.
Thus it is enough to show, inductively, that is a bijection. We must show that an independent set
is not in
if and only if
if and only if
, clearly
if and only if (
Observe that this is essentially a generalisation of Koh’s Theorem 6.2. Indeed, as a tight embedding of . As Dilworth’s decomposition theorem assures there is such a decomposition when
We continue, and complement Theorem 6.2 by finding every poset . As per Remark 6.5, this is done by finding every acyclic transitive graph
. The main elements required are anti-chain analogues of Corollary 4.2 and Lemma 4.3. Instead of ‘repeating’ the proofs of these earlier sections, we observe that the anti-chain analogues follow from the results of Section 4 by comparing the constructions of
6.4. Properties of A ℘(L ) via Koh’s Construction
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
Comparing definitions, or comparing Figure 2 to Figure 4, it is easy to see how to get , let
with arcset
One can easily check that , and more generally that
It is a simple matter to show the transitivity of , (i.e. tournament decompositions of
. As he considered only posets, this difference would be cosmetic for him, but it is significant for us. The construction can be used to easily translate statments about
Using Corollary 4.2, we get the following analogue.
Corollary 6.9
.
The classification lemma, Lemma 4.3, yields the following analogue.
Lemma 6.10
L is a {0, 1}-sublattice if and only if the minimum and maximum element of each tournament T in are loopless in A. L is a subdirect sublattice if and only if A is loopless, which holds if and only if contains no directed cycles. L is a tight sublattice if and only if A contains no directed cycle or alternating cycle: v 1 →u 1 ←v 2 →u 2 ← ·· ·→uℓ ←v 1.
For tight sublattices
6.5. Complement to Koh
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
For a lattice . This gives, by Fact 5.2, a pointed chain decomposition
of some preorder
, we can apply construction
. By Theorem 6.8 we have
. In particular, if the embedding
As a complement to Koh’s theorem, we want to show that for any poset , there is some
Corollary 6.11
Let . The main task is to find a tournament decomposition
of
Where into
, we alter
such that
Corollary 6.12
Acknowledgements
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
I thank the anonymous referees for their patience and their constructive comments.
Figures
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
References
- Abstract
- 1. Introduction
- 2. Notation and Background
- 3. The Rival Construction
D ℘(L ) - 3.3. Some Examples
- 3.4. The Closure of ℐ and Transitivity of
D ℘(ℐ) - 4. Main Results
- 4.2. One-to-one Correspondence
- 4.3. Classification of Sublattices
- 5. Back to Embeddings
- 5.2. The Full Correspondence Extending Corollary 2.5
- 6. Lattices of Independent Sets
- 6.3. The Alternate Rival Construction
A ℘(L ) - 6.4. Properties of
A ℘(L ) via Koh’s Construction - 6.5. Complement to Koh
- Acknowledgements
- Figures
- References
- G. Birkhoff.
Rings of sets . Duke Math J.,3 (3)(1937), 443-454. - R. Dilworth.
A decomposition theorem for partially ordered sets . Ann of Math.,51 (1950), 161-166. - R. Dilworth.
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