### 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 _{L}

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

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,

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 _{℘}(_{℘}(

This preorder is an analogue of _{℘}(

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

### 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 _{L}_{L′}_{L}_{L′}_{x}_{∈}_{S} x_{L} S_{n}_{+1} on the _{1}, . . . , _{d}_{1}, . . . , _{d}_{i}_{i}_{i}_{i}

In his famous representation theorem in [1], Birkhoff showed that where , for a poset

### Theorem 2.1.([1])

_{L} S.

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

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 _{I}_{∈ℐ}

**Theorem 3.1.([8])**

### 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 _{i}_{i}_{i}_{C}_{i}. The subscript differentiates it from the element 1 that is occasionally used in examples to label the first or second element of a chain. We let _{C}_{1}, . . . , 0_{C}_{d}) and _{C}_{1}, . . . , 1_{C}_{d}) denote the zero and unit of ℘. When _{L}_{L}

for _{i}

for _{j}_{i}^{j}_{i}, β^{j}_{i}_{j}_{i}, β_{i}

Given and , let be the subposet of ℘ induced by _{i}_{C}_{i}. This is consistant with viewing _{i}_{j}_{C}_{j}. These shortcuts will help us avoid having to deal seperately with the extremal cases of

Recall that a poset, and in particular , is a reflexive relation. We refer to a reflexive relation simply as a

The following is our extension of

**Definition 3.2. (**D D L ))

_{℘}(ℐ) and_{℘}(For any family of chains and any family of irreducible intervals ℐ of , let _{℘}(ℐ) be the spanned extension of that we get from by letting (_{j}_{i}_{i}, β^{j}_{℘}(_{℘}(ℐ_{L}

An (_{1} → _{2} → · · · → _{ℓ}_{1}, _{2}, . . . , _{ℓ}_{i}_{i}_{+1} for each

**Definition 3.3**

A subset

**Definition 3.4. ( and )**

For any digraph

As with posets, it is clear for digraphs that unions and intersections of downsets are downsets, so is indeed a poset for any digraph _{℘}(

For a poset ^{∨} be the poset we get from ^{⋄} be the poset we get from ^{∨} by adding a new maximum element. It has been observed in several places, (see, for example, [6]), that

Indeed, the isomorphism is the map that drops the appended unit. As implies

With

**Lemma 3.5**

_{℘}

**Proof**

Indeed ℘ is a lattice, so we may apply Theorem 2.1, and then

Before we prove our main theorem, that _{L}_{℘}(

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 ℘ = _{5}×_{6} and _{1} = ℘[3_{2}, 2^{1}] as are shown in Figure 1, but now, instead of _{1}), it shows with them the digraph and its spanned extension _{℘}(_{1}).

The construction of _{℘}(_{i}, β^{j}_{2}, 2^{1}] from ℘, we see that we should add the 3_{2} ≥ 3_{1} to to maintain the isomorphism. Indeed, this ensures that, for example, the set _{1}, 1_{1}, 0, 1_{2}, 2_{2}, 3_{2}} is no longer a downset. In general it ‘kills’ any downset _{2} but not 3_{1}. These are exactly the downsets _{2}, 2^{1}].

The third example in the figure shows a non-tight sublattice _{2} of ℘, observe that the corresponding extension _{℘}(_{2}) of 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 _{℘}(ℐ) is closed if and only if ℐ is transitive. It is a pretty intuitive notion, but we prove it via a technical lemma that we be quite useful later.

Recall that we included the empty intervals of the form [_{i}, β^{i}_{℘}(ℐ) containing . In particular, we assume for any sublattice _{L}_{L}

Now our technical lemma.

### Lemma 3.6

_{℘}(ℐ)

[

α ] ⊆ ∪ℐ._{i}, β^{j}For all x ∈L, x ≥_{i}α ⇒x _{j}> β.There is a ((β + 1) )_{j}, α_{i}-path in D.

**Proof**

The equivalence of (i) and (ii) is immediate as both are equivalent to the statement that there are no elements _{i}_{j}

On the one hand, consider an ((_{s}_{ℓ}, (_{1})_{s}_{1})-path in

By the definition of _{i}_{s}_{i}, (_{i}_{+1} − 1)^{s}^{i+1}] ∈ ℐ for each _{s}_{1} ≥ _{1}. By the equivalence of (i) and (ii) this implies _{s}_{2}_{2} − 1, which in turn implies _{s}_{3}_{3} − 1, etc., until we get that _{s}_{ℓ}_{ℓ}

On the other hand, assume that there is no such path from (_{ℓ}_{s}_{ℓ} to (_{1})_{s}_{1}. We will find _{s}_{1} ≥ _{1} and _{s}_{ℓ}_{ℓ}_{k}_{k}_{k}_{1})_{s}_{1}. If no such path exists, let _{k}_{s}_{1} ≥ _{1}, and by the assumption that there is no path from (_{ℓ}_{s}_{ℓ} to (_{1})_{s}_{1} we have that _{s}_{ℓ}_{ℓ}

If _{i},^{j}_{i},^{j}_{j}_{i}_{i},^{j}_{i}_{j}_{i}_{i}_{1})_{s}_{1}, so the arc ((_{j},_{i}_{j}_{s}_{j} to (_{1})_{s}_{1}. But then _{j}_{j}_{j}_{j}

As ℐ_{L}_{℘}(

### Proposition 3.7

_{℘}(ℐ)

**Proof**

On the one hand, assume that ℐ is closed. Then we can replace (i) of Lemma 3.6 with [_{i}, β^{j}_{k}_{j}_{j}_{i}_{i}, γ^{k}_{k}_{i}

On the other hand, assume that ℐ is not closed. Then there is some [_{i}, β^{j}_{i}, β^{j}_{i}, β^{j}

### 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.

**Theorem 4.1**

_{℘}(

_{℘}

**Proof**

As it simplifies induction, we prove the slightly more general result that _{℘}(ℐ) where ℐ is any family of irreducible intervals of ℘ such that

In the general case, we first observe that is a subposet of . Indeed, as

Now it is enough to show that a bijection. We do this by induction on the size of ℐ. Let [_{i}, β^{j}_{i}, β^{j}_{℘}(ℐ′). By induction we have that is a bijection.

We must show for , that if and only if _{i}, β^{j}_{j}, α_{i}_{i}_{j}_{i}_{j}_{i}, β^{j}

To see that ^{−1} = _{℘}

### 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

is such that is the sublattice ℘ ∪ℐ_{D}

### 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 _{i}_{i}

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 _{3} × _{3} ([2_{1}, 1^{2}] ∪ [2_{2}, 1^{1}]) shown in Figure 3 is a subdirect sublattice of _{3} × _{3}, but not tight. One notices in this example that _{℘}(

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 α → (_{i}α − 1)_{i}.L is tight if and only if D is a poset.

**Proof**

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 if1 is a sink. The result follows.We have

α → (_{i}α − 1) in_{i}D if and only if there is no downsetT (x ) ofD containing (α −1) but not_{i}α . This is if and only if there is no element_{i}x ∈L such thatx =_{i}α − 1.First assume that

L is tight, and assume, towards contradiction thatD contains a cycle (α _{1})_{i}_{1}→ (α _{2})_{i}_{2}· · · → (α )_{ℓ}_{i}_{ℓ}→ (α _{1})_{i}_{1}, for someℓ ≥ 2. We show that no vertexx inL hasx _{i}_{1}=α _{1}, which contradicts the fact thatL is tight. Indeed, ifx did havex _{i}_{1}=α _{1}, thenx _{i}_{1}≥α _{1}so by Lemma 3.6x _{i}_{2}> α _{2}sox _{i}_{3}> α _{3}etc., and we get thatx _{i}_{1}> α _{1}, a contradiction.

On the other hand, assume that

### Remark 4.4

Though our goal was a correspondence including

For any preorder ^{*} be the subgraph induced by

is a lattice isomorphism. When _{℘}(_{℘}(^{*}, is a poset. It follows that _{℘}(^{*} =

Observing that 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 _{℘}(

A

### 5.1. The Canonical Quotient of D _{℘}(L )

The _{~} is well known to be a poset. Classes of ^{D}^{D̃}_{[}_{x}_{]∈}_{S}

**Lemma 5.1**

^{⋄}(

**Proof**

As [

As , the second fact follows by

Now for any pointed chain decomposition of a preorder

Thus we have the following.

**Fact 5.2**

### 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 ^{⋄} = ^{⋄}(_{℘}(^{⋄}.

Using the isomorphism that we get from Theorem 4.1, we will see that we can write _{e}

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 _{⋄}. By Lemma 5.1 we have and Theorem 4.1 gives us an embedding

of ^{−1}(

of

We are ready for our generalisation of Corollary 2.5. Recall that a pointed chain cover of ^{⋄}(^{⋄}(

### Theorem 5.3

_{e} and φ_{φ} defined in equations^{⋄}(

**Proof**

It is enough to show that the maps _{e}_{φ}^{⋄} we have that _{℘} is in 〈_{℘} for some other _{φ}_{e} ; indeed, recalling for the last equality that and

To see that _{e}_{φ} = ^{⋄}. Using in the third line 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

**Theorem 6.1.([3])**

In [5], Koh showed a converse: every finite distributive lattice

**Theorem 6.2.([5])**

There are posets

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 _{1} → _{2} → ·· ·→_{d}

**Definition 6.3**

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.

**Definition 6.4.( )**

Let be the family of independent sets of size

**Remark 6.5**

For a poset

Our basic setup now is as follows. Given a product of a family of _{i}_{j}_{i}^{(}^{i}^{)} or _{1}, . . . , _{d}_{x}

By Remark 6.5 the _{x}

A digraph _{x}

**Lemma 6.6**

_{x}, I_{y}_{x}_{y} if and only x_{i}_{i} for all i.

**Proof**

The ‘if’ direction is immediate from Definition 6.4 as all arcs of are in _{x}_{y}_{i}_{i}_{i} < x_{i}_{i}_{i}_{x}_{y}_{i}_{j}_{i}_{i}_{i}_{i}_{i}_{i}_{j}_{i}_{j}

From here, given a digraph

### 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 _{℘}(ℐ) be the spanned extension of that we get from by letting ^{(}^{j}^{)} → ^{(}^{i}^{)} for each [_{i}, β^{j}_{℘}(_{℘}(ℐ_{L}

See Figure 4 for examples of _{℘}(_{℘}(_{3} = ℘ [3_{2}, 4^{1}], we have both of the arcs (4^{(1)}, 3^{(2)}) and (5^{(2)}, 3^{(2)}), as the intervals [3_{2}, 4^{1}] and [3_{2}, 5^{2}] are the same. The graphs in Figure 4 are all irreflexive except _{℘}(_{3}) which, because it is transitive, has loops on the vertices 3^{(2)}, 4^{(2)}, and 5^{(3)}.

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 [3_{1}, 3^{2}] as we do to get the second lattice _{2} = _{1} [3_{1}, 3^{2}] of Figure 2. The second figure of Figure 4 shows what we must add to . To ensure that, for example, _{(2,3)} = {2^{(1)}, 3^{(2)}} is no longer independent, we add the arc (2^{(1)}, 3^{(2)}) to . This also ensures that such sets as _{(2,4)} and _{(1,5)} are no longer independent.

### Theorem 6.8

_{℘}(

**Proof**

The theorem holds by Lemma 6.6 in the simplest case where

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 _{i}, β^{j}_{i}_{j}^{(}^{j}^{)}, ^{(}^{i}^{)}) completes a path between _{j}_{i}_{i}_{j}

Observe that this is essentially a generalisation of Koh’s Theorem 6.2. Indeed, as a tight embedding of

We continue, and complement Theorem 6.2 by finding every poset _{℘}(_{℘}(

### 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 _{℘}(_{℘}(_{℘}(

One can easily check that , and more generally that _{℘}(_{℘}(ℐ)) = _{℘}(ℐ) for any closed family ℐ of irreducible intervals. (We require that ℐ is closed for this as an out-arc of _{℘}(

It is a simple matter to show the transitivity of _{℘}(_{℘}(ℐ) is transitive if ℐ is closed. Observing that the construction _{℘}(_{℘}(_{℘}(

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}← ·· ·→u ←_{ℓ}v _{1}.

For tight sublattices _{℘}(^{⋄} (_{℘}(^{⋄}). An important part of Koh’s proof from [5] was that his version of _{℘}(^{⋄}) contains no crown, which is essentially what item (iii) says.

### 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 _{e}_{℘}(_{e}_{e}_{℘}(^{⋄} (_{e}_{℘}(_{e}_{e}

As a complement to Koh’s theorem, we want to show that for any poset _{e}

### Corollary 6.11

_{e}.

**Proof**

Let

Where ^{r}^{r}^{−1}(_{e}_{φ} where _{φ}

### 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.
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, Birkhäuser/Springer Basel AG, Basel, 2011. - KM. Koh.
On the lattice of maximum-sized antichains of a finite poset . Alg Universalis.,17 (1983), 73-86. - RN. Larson.
Embeddings of finite distributive lattices into products of chains . Semi-group Forum.,56 (1998), 70-77. - V. Retakh, and M. Saks.
On the rational relationships among pseudo-roots of a non-commutative polynomial .,Preprint. - I. Rival.
Maximal sublattices of finite distributive lattices II . Proc Amer Math Soc.,44 (1974), 263-268. - M. Siggers.
Distributive lattice polymorphisms on reflexive graphs . Bull Korean Math Soc.,55 (2018), 81-105.