Interpolation of monotone functions in lattices

Martin Goldstern, TU Wien

October 20, 1997

Abstract We show that for every lattice L_{0} and for every cardinal κ there is a lattice L ⊇ L_{0} on which every monotone function can be interpolated by a polynomial on any set of size κ.

Introduction

Let (L, ∧, ∨) be a lattice. The polynomials over L are the well-formed expressions involving constants from L and/or formal variables x_{1}, x_{2}, … together with parentheses and the formal symbols ∧, ∨. The set of n-ary polynomial functions is the smallest set of functions from L^{n} to L which contains all the constant functions as well as the projection functions and is closed under (pointwise) meet and join. Every polynomial in which at most the variables x_{1}, …, x_{n} occur defines in an obvious way an n-ary polynomial function. Clearly every polynomial function f is monotone.

We call a lattice L n-order polynomially complete if

(∗) every monotone n-ary function is a polynomial function

and we say that L is order polynomially complete if L is n-order polynomially complete for every n.

The finite order polynomially complete lattices were investigated and classified by Schweigert [7], Wille [8], Kindermann [6] and others. The question whether infinite order polynomially complete lattices exist is still open. In [5], Kaiser and Sauer showed (a) Every order polynomially complete lattice is bounded (i.e., has a largest

and a smallest element)

(b) Every infinite lattice contains an infinite chain or an infinite antichain

(c) If L contains an antichain of size κ or a chain of order type κ + 1 or (κ + 1)^{∗}, then L admits at least 2^{κ} many monotone unary functions,

and concluded that an infinite 1-order polynomially complete lattice (if one exists at all) cannot be countable, but must have size at least continuum. Note that (b) and (c) above are true not only for lattices but for arbitrary partial orders, so the theorem is only partially of an “algebraic” nature. The search for a algebraic proof of the conjecture “there is no order polynomially complete infinite lattice” naturally leads to the following problem:

Find a “structural” property of polynomial functions and show that in every infinite lattice there is a monotone function not satisfying this property.

In this paper we show that there are no “local” properties of polynomial functions which are not shared by all monotone functions. Formally, we prove:

Theorem A 0.1. Let L_{0} be a lattice, n a natural number, f_{0} : L_{0}^{n} → L_{0} a monotone partial function. Then there is a lattice L which contains L_{0} as a sublattice and a polynomial function f on L which extends f_{0}.

Thus, any monotone function on a lattice can be realized as the behaviour of a polynomial on a suitable extension lattice. Is it then possible to represent ALL monotone functions by polynomials? We can give a “local” answer to this question in the next theorem. Noticing that the lattice L in theorem A can be chosen to be of the same cardinality as L_{0}, we can iterate theorem A to obtain:

Theorem B 0.2. Let λ and κ be infinite cardinals, and let L_{0} be a lattice of size ≤ λ^{κ}. Then there is a lattice L of size λ^{κ}, L_{0} ⊆ L such that for every n, for every partial monotone function f : L^{n} → L with domain of size ≤κ there is a polynomial function f′ on L which extends f, i.e., L is “κ-locally order polynomially complete.”

In particular, for any lattice L_{0} we can (choosing κ = λ = |L_{0}|) get a lattice L ⊇ L_{0} of size 2^{|L_{0}|} such that every monotone function f : L_{0} → L_{0} is induced by a polynomial over L. (In the terminology of [4] we could phrase this as follows: Every lattice is “extension order polynomially complete.”)

For example, letting λ = κ = ℵ_{0}, we get a lattice of size continuum on which every monotone function can be interpolated by a polynomial on every countable set.

Notation 0.3. Lattices are denoted by L, L′, L_{1}, etc. When we consider several lattices, we use the self-explanatory notation ∧_{1}, ≤_{2}, etc. for the operations/relation in L_{1}, L_{2}, etc. We read lattice operations from left to right: a ∨ b ∧ c = (a ∨ b) ∧ c, a ∧ b ∨ c = (a ∧ b) ∨ c. For a ∈ L we let (a] := {b ∈ L : b ≤ a}.

L_{1} ⊆ L_{2} or L_{2} ⊇ L_{1} always means that L_{1} is a sublattice of L_{2}.

We use α, β, γ to range over ordinals, κ, λ, μ for cardinals. The set of natural numbers is denoted by ω. |A| is the cardinality of the set A. λ^{κ} (cardinal exponentiation) is the cardinality of {f : f : κ → λ}.

f, g, h are always unary functions, dom(f) is the domain of f and ran(f) is the range of f. f : L_{1} → L_{2} means dom(f) = L_{1}, ran(f) ⊆ L_{2}, unless we explicitly call f “partial”, in which case we only have dom(f) ⊆ L_{1}.

f↾A is the restriction of f to A. “f extends g” or “g ⊆ f” means g = f↾dom(g). f[A] = ran(f↾A).

We write _{⌣} · ·^{}_{x} to denote the end of the proof of theorem (lemma, etc.) x, and we write – · ·^{}_{y} if (most of) a proof is left to the reader. _{⌢} · ·^{}marks the end of a wrong proof.

Acknowledgements 0.4. I am grateful to Hans Kaiser for providing the initial motivation, and to Gerhard Dorfer for simplifying the original proof in section 2.

1 Continuous join-homomorphisms

First we remark that it is enough to prove theorem A for the case n = 1. This follows easily from fact 1.2 below:

Definition and Fact 1.1. Let L be a lattice, n > 0. Define δ_{n} : L → L^{n} by δ_{n}(x) = ⟨x, …, x⟩. Then δ_{n} is an embedding of lattices.

Fact 1.2. Let L be a bounded lattice, n > 0. Then there is a polynomial π_{n} over L^{n} such that the induced function from (L^{n})^{n} → L^{n} (which we again call π_{n}) satisfies

π_{n}(δ_{n}(a_{1}), …, δ_{n}(a_{n})) = ⟨a_{1}, …, a_{n}⟩

for all a_{1}, …, a_{n} ∈ L.

Proof: Let a = inf L, b = sup L. Let e_{1} := ⟨b, a, …, a⟩, …, e_{n} := ⟨a, …, a, b⟩. Let π_{n} be the polynomial (x_{1} ∧ b_{1}) ∨ ⋯ ∨ (x_{n} ∧ b_{n}). _{⌣} · ·^{}_{1.2}

Conclusion 1.3. It is enough to prove theorem A for the case n = 1.

Proof: Without loss of generality assume that L_{0} is bounded. Let f_{0} be an n-ary function on L_{0}. So δ_{n} ∘ f_{0} is a unary function on L_{0}^{n}. Find a lattice L ⊇ L_{0}^{n} and a polynomial function f′ : L → L extending δ_{n} ∘ f_{0}. Now let f : L^{n} → L be defined by f(x_{1}, …, x_{n}) = f′(π_{n}(x_{1}, …, x_{n})). It is clear that (identifying L_{0} with δ_{n}[L_{0}] ⊆ L_{0}^{n} ⊆ L) f is a polynomial function extending f_{0}. _{⌣} · ·^{}_{1.3}

So from now on we will only on concentrate on unary functions.

Definition 1.4. A lattice L is complete, if every subset A of L has a greatest lower bound, called inf A. In a complete lattice, also every set has a least upper bound, sup A. In particular, L must be bounded: 0 = inf L = sup ∅, 1 = inf ∅ = sup L.

Fact 1.5. For every lattice L there is a complete lattice L̄ such that L ⊆ L̄.

Proof: Well known (see, e.g., [3]). Let L̄ be the set of ideals of L, i.e., the set of downward closed upward directed subsets of L (including the empty set), and embed L via x ↦ {y : y ≤ x}. – · ·^{}_{1.5} Fact 1.6. (a) If f : L_{1} → L_{2} is a partial monotone function, and if L_{2} is a complete lattice, then f can be extended to a total monotone function.

(b) If f is a partial monotone function from L to L then there is a lattice L̄ of the same cardinality as L and a total monotone function f̄ : L̄ → L̄ extending f. Proof: (a): Let f̄(x) = sup{f(y) : y ≤ x, y ∈ dom(f)}. (b): Let L′ ⊇ L be a complete lattice, and let f′ : L′ → L′ be a total function extending f. Now let L̄ ⊆ L′ be the structure generated from L by the binary operations ∧ and ∨ and the unary function f′. _{⌣} · ·^{}_{1.6}

To explain the motivation for the following definition, consider a complete lattice L and the polynomial p(x) = x ∨ a for some fixed a ∈ L. Clearly p(x∨y) = p(x)∨p(y) and moreover p(sup A) = (sup A)∨a = sup(A∪{a}) = sup{x ∨ a : x ∈ A} = sup p[A] for any nonempty A ⊆ L. Definition 1.7. f : L_{1} → L_{2} is called join-homomorphism if for all x, y ∈ L_{1} we have f(x) ∨ f(y) = f(x ∨ y). We call f : L_{1} → L_{2} a continuous join-homomorphism if

(a) f is monotone

and there is a function f^{∗} : L_{2} → L_{1} such that

(b) ∀y ∈ L_{2} f(f^{∗}(y)) ≤ y

(c) ∀x ∈ L_{1} ∀y ∈ L_{2} f(x) ≤ b ⇒ a ≤ f^{∗}(y).

We define “meet-homomorphism” and “continuous meet-homomorphism” dually. Remark 1.8. 1. Every join homomorphism is monotone.

2. Every continuous join-homomorphism is a join-homomorphism.

3. If f is a continuous join-homomorphism, then f^{∗} is unique, namely, f^{∗}(y) must be the greatest element of

{x ∈ L_{1} : f(x) ≤ y}

4. (a)&(b)&(c) above can be replaced by (a)&(bc), where

(bc) ∀a ∈ L_{1} ∀b ∈ L_{2} f(a) ≤ b ⇔ a ≤ f^{∗}(b).

5. If L_{1} is a complete lattice, then f : L_{1} → L_{2} is a continuous joinhomomorphism iff f(sup A) = sup f[A] for all A ⊆ L_{1}.

Proof: (1), (3) and (4) are clear.

(2): Since f is monotone, f(x ∨ y) ≥ f(x) ∨ f(y). Conversely, let z := f(x) ∨ f(y). Then f(x) ≤ z, so x ≤ f^{∗}(z), and similarly y ≤ f^{∗}(z). Hence x ∨ y ≤ f^{∗}(z), so f(x ∨ y) ≤ ff^{∗}(z) ≤ z = f(x) ∨ f(y) and we are done.

(5): If f(sup A) = sup f[A], then f is clearly a join homomorphism, hence monotone. Let f^{∗}(y) := sup{x : f(x) ≤ y}. It is clear that ff^{∗}(y) ≤ y, and that f(x) ≤ y implies x ≤ f^{∗}(y).

Conversely, assume that f is a continuous join-homomorphism. Let A ⊆ L_{1}, and let a = sup A. The monotonicity of f shows that f(a) is an upper bound for f[A]. Now let b be any upper bound for f[A]. We have ∀x ∈ A f(x) ≤ b, hence ∀x ∈ A x ≤ f^{∗}(b), so a = sup A ≤ f^{∗}(b), hence f(a) ≤ f(f^{∗}(b)) ≤ b. So f(a) is indeed the least upper bound of f[A].

⌣ · ·^{}_{1.8}

The next lemma shows that the property isolated in definition 1.7 more or less characterizes the possible behaviours of the polynomial function x ↦ x ∨ a, at least for complete lattices.

Definition 1.9. Let L_{1} and L_{2} be disjoint lattices, f : L_{1} → L_{2}. We define L_{1} +_{f} L_{2} = (L_{1} ∪ L_{2}, ≤) as follows:

x ≤ y iff: either x, y ∈ L_{1}, x ≤_{1} y
or x, y ∈ L_{2}, x ≤_{2} y
or x ∈ L_{1}, y ∈ L_{2}, f(x) ≤_{2} y

[figure: f; x; f(x)]

Lemma 1.10. Assume that f : L_{1} → L_{2} is a continuous join-homomorphism. Let L := L_{1} +_{f} L_{2}. Then

(a) L is a lattice.

(b) L_{1}, L_{2} ⊆ L.

(c) If L_{2} has a smallest element a_{2}, then for all x ∈ L_{1} we have f(x) = x∨a_{2}.

Proof: Transitivity of ≤ follows from the monotonicity of f: Let x ≤ y ≤ z. The only nontrivial case is when x ≤_{1} y and f(y) ≤_{2} z. But then also f(x) ≤_{2} f(y) ≤_{2} z.

Now let f^{∗} : L_{2} → L_{1} be such that f^{∗}(y) is always the greatest element of {x ∈ L_{1} : f(x) ≤ y}. We claim that for x ∈ L_{1}, y ∈ L_{2} we have inf{x, y} = x ∧_{1} f^{∗}(y). For a proof, note that z is a lower bound for {x, y} iff z ≤_{1} x and f(z) ≤_{2} y, which is equivalent to z ≤_{1} x and z ≤_{1} f^{∗}(y), i.e., z ≤ x ∧_{1} f^{∗}(y).

We leave it to the reader to check the other cases of the following table:

inf\{x, y\} = \{ \begin{matrix}x ∧_{1} y & if x, y ∈ L_{1} \\ x ∧_{2} y & if x, y ∈ L_{2} \\ x ∧_{1} f^{∗}(y) & if x ∈ L_{1}, y ∈ L_{2} \end{matrix}

sup\{x, y\} = \{ \begin{matrix}x ∨_{1} y & if x, y ∈ L_{1} \\ x ∨_{2} y & if x, y ∈ L_{2} \\ f(x) ∨_{2} y & if x ∈ L_{1}, y ∈ L_{2} \end{matrix}

(b) and (c) are now clear. _{⌣} · ·^{}_{1.10}

Conversely, it is easy to check the following: If f : L_{1} → L_{2}, and L_{1} has a largest element b_{1}, then we also have:

If L_{1}+_{f} L_{2} is a lattice, then f is a continuous join-homomorphism, witnessed by f^{∗}(y) = y ∧ b_{1}.

The construction in lemma 1.10 deals with the situation where the function which we want to represent by a polynomial has disjoint domain and range. Below we give a slight extension of the previous lemma that deals with the case f : L → L. (The following lemma is also a generalization of a construction considered in [2].)

Lemma 1.11. Let L_{0}, L_{1}, …, L_{5} be disjoint lattices, and let f_{i} : L_{i} → L_{i+1}, i = 0, …, 5 be monotone functions. Assume that each f_{i} is a continuous meet-homomorphism for i = 0, 2, 4 and a continuous join-homomorphism for i = 1, 3, 5. Then the following structure L is a lattice:

[figure: L_{4}; L_{5}; L_{0}; L_{1}; L_{2}; L_{3}; 1; 0; f_{0}; f_{1}; f_{2}; f_{3}; f_{4}; f_{5}]

The universe of L is the disjoint union L_{0} ∪ L_{1} ∪ ⋯ ∪ L_{5} ∪ {0, 1}. The partial order on L is defined as follows:

8

x ≤ y iff x = 0 or y = 1
or for some i ∈ {0, …, 5}, x, y ∈ L_{i} and x ≤_{i} y

or for some i ∈ {1, 3, 5}, x ∈ L_{i}, y ∈ L_{i+1} and f_{i}(x) ≤_{i+1} y or for some i ∈ {0, 2, 4}, x ∈ L_{i+1}, y ∈ L_{i} and x ≤_{i+1} f_{i}(y).

Moreover, if a_{i} = inf L_{i}, b_{i} = sup L = i, then for all x ∈ L_{0} we have

f_{5} f_{4} f_{3} f_{2} f_{1} f_{0}(x) = x ∧ b_{1} ∨ a_{2} ∧ b_{3} ∨ a_{4} ∧ b_{5} ∨ a_{0}.

(It is understood that addition of indices is performed modulo 6, i.e. L_{6} = L_{0}, L_{7} = L_{1}, etc.) Proof: We leave it to the reader to check that join and meet can be computed accoding to the following rules:

• inf{x, 0} = 0, sup{x, 0} = inf{x, 1} = x, sup{x, 1} = 1.

• If x, y ∈ L_{i}, then inf{x, y} = x ∧_{i} y, sup{x, y} = x ∨_{i} y.

• If x ∈ L_{i}, y ∈ L_{i+1}, i even, then inf{x, y} = f_{i}(x) ∧_{i+1} y, sup{x, y} = x ∨_{i} f_{i}^{∗}(y).

• If x ∈ L_{i}, y ∈ L_{i+1}, i odd, then inf{x, y} = x ∧_{i} f_{i}^{∗}(y), sup{x, y} = f_{i}(x) ∨_{i+1} y.

• If x ∈ L_{i}, y ∈ L_{i+2}, i odd, then inf{x, y} = 0, sup{x, y} = f_{i}(x) ∨_{i+1} f_{i+1}^{∗}(y).

• If x ∈ L_{i}, y ∈ L_{i+2}, i even, then inf{x, y} = f_{i}(x) ∧_{i+1} f_{i+1}^{∗}(y), sup{x, y} = 1.

• If x ∈ L_{i}, y ∈ L_{i+3}, then inf{x, y} = 0, sup{x, y} = 1.

– · ·^{}_{1.11}

2 Factoring monotone functions

By what we have seen so far, theorem A is true if we change the condition “f monotone” to “f a continuous join-homomorphism”. The dual is of course also true. Our next aim is to show that every monotone function can be written as a composition of a continuous join-homomorphism and a continuous meet-homomorphism.

Definition 2.1. For any lattice L let L̂ be the set of all downward closed subsets of L:

L̂ = {A ⊆ L : ∀a ∈ A ∀b ≤ a b ∈ A}.

Fact 2.2. For any lattice L the partial order (L̂, ⊆) is a complete lattice, with operations

A ∨ B = A ∪ B (more generally, sup A = ⋃ A for any A ⊆ L̂)
A ∧ B = {z : ∃x ∈ A ∃y ∈ B z ≤ x ∧ y}

Proof: Straightforward. – · ·^{}_{2.2}

Lemma 2.3. Assume that L is a complete lattice. Then the map g : L → L̂, defined by g(x) = (x] := {y ∈ L : y ≤ x}, is a continuous meethomomorphism.

Proof: We will use the dual of remark 1.8(5). x ≤ y implies (x] ⊆ (y], hence g is monotone. Let A ⊆ L, a = inf A. We have to show g(a) = inf g[A]. It is clear that g(a) is a lower bound for g[A], since g is monotone. So let B ∈ L̂ be any lower bound for g[A]. Then B ⊆ (x] for all x ∈ A, hence each b ∈ B is a lower bound for A. So each b ∈ B is ≤ a, so B ⊆ (a] = g(a). _{⌣} · ·^{}_{2.3}

Factor Lemma 2.4. Assume that f : L_{0} → L_{2} is monotone, and L_{2} is complete. Then there is a function h : L̂_{0} → L_{2} satisfying • h is a continuous join-homomorphism

• h ∘ g = f (where g is the continuous meet-homomorphism from the previous lemma)

L_{0} \underrightarrow{f} L_{2} ; g ↘ ↗ h ; L̂_{0}

Proof: Let h(A) = sup f[A]. Clearly h is monotone. Also, since f is monotone we get

h(g(x)) = sup f[{y : y ≤ x}] = f(x)

For any A ⊆ L̂ we have sup A = ⋃ A and

sup h[A] = sup{sup{f(x) : x ∈ A} : A ∈ A} = sup{f(x) : x ∈ ⋃ A} = h(sup A).

⌣ · ·^{}_{2.4}

Now we can put the pieces together: Proof of theorem A: By conclusion 1.3 we only have to consider the case of a unary function. Let L_{0} be a lattice, f : L_{0} → L_{0} a partial monotone function. By facts 1.5 and 1.6 we may assume that L_{0} is complete, and that f is a total function. Let L_{1} := L̂_{0} as in definition 2.1, and let f_{0} : L_{0} → L_{1} be the continuous meet-homomorphism defined in lemma 2.3, f_{0}(x) = (x]. Let L_{2} be an isomorphic copy of L_{0}, i : L_{0} → L_{2} an isomorphism. The map i∘f : L_{0} → L_{2} is monotone, so we can find a continuous join-homomorphism f_{1} : L_{1} → L_{2} such that f_{1} ∘ f_{0} = i ∘ f.

\begin{matrix}L_{0} & \underrightarrow{i ∘ f} & L_{2} \\ f_{0} ↘ & ↗ f_{1} \\ L_{1} \end{matrix}

Let L_{3}, L_{4}, L_{5} be disjoint isomorphic copies of L_{2} with isomorphisms

L_{2}  L_{4}  L_{0}
f_{2}↘  f_{3}↗  f_{4}↘  f_{5}↗
L_{3}  L_{5}

such that f_{5} ∘ f_{4} ∘ f_{3} ∘ f_{2} ∘ i : L_{0} → L_{0} is the identity map on L_{0}. Then f_{5} ∘ f_{4} ∘ f_{3} ∘ f_{2} ∘ f_{1} ∘ f_{0} = f : L_{0} → L_{0}.

Letting L := L_{0} ∪ ⋯ ∪ L_{5} ∪ {0, 1} as in lemma 1.11, we get a lattice L. Letting a_{i} := inf(L_{i}), b_{i} := sup(L_{i}) we get

∀x ∈ L_{0} f(x) = x ∧ b_{1} ∨ a_{2} ∧ b_{3} ∨ a_{4} ∧ b_{5} ∨ a_{0}
⌣ · ·^{}_{0.1}

Remark 2.5. We may assume that the lattice L is of the same cardinality as L_{0}. (Assuming, of course, that L_{0} is infinite. If L_{0} is finite the L will also be finite of size ≤ 2^{|L_{0}|} + 5|L_{0}| + 2.)

Proof: Let L′ be the sublattice of L generated by the elements of the original (possibly non-complete) L_{0} plus the coefficients of the polynomial p.

⌣ · ·^{}_{2.5}

3 Proof of theorem B

Let μ := λ^{κ}. Let Γ : μ × μ × ω → μ be a bijection with the property

(∗∗)∀β, γ < μ ∀n : Γ(β, γ, n) ≥ γ

Start with an arbitrary infinite lattice L_{0} of cardinality ≤ μ. We will construct a continuous increasing sequence (L_{α} : α ≤ μ) of lattices and a sequence (p_{α} : α < μ) of polynomials, where p_{α} has coefficients in L_{α+1}, and |L_{α}| ≤ max(|α|, |L_{0}|).

First we deal with successor steps: L_{α} has size ≤ μ so for any n the set of partial functions from L_{α}^{n} to L_{α} whose domain has size ≤ κ has cardinality at most μ^{κ} = μ. Let (f_{α,n}^{β} : β < μ) be an enumeration of these functions. We can find β_{α}, γ_{α} ∈ μ, n_{α} ∈ ω such that Γ(β_{α}, γ_{α}, n_{α}) = α. As γ_{α} ≤ α, we already know what f_{γ_{α},n_{α}}^{β_{α}} is, namely, a partial function from L_{γ_{α}}^{n_{α}} to L_{γ_{α}}, hence also a partial function from L_{α}^{n_{α}} to L_{α}.

By theorem A we can find a lattice L_{α+1} ⊇ L_{α} of again the same size and a polynomial p_{α} with coefficients in L_{α+1} such that the polynomial function induced by p_{α} (which we again call p_{α}) extends f_{α} and hence also f_{γ_{α},n_{α}}^{β_{α}}.

This concludes the construction of L_{α+1} and p_{α}.

If α is a limit ordinal, we let L_{α} := ⋃_{β<α} L_{β}.

Finally, let L := L_{μ} = ⋃_{α<μ} L_{α}. L is of size μ. Let f be an arbitrary partial monotone function from L^{n} to L with domain of cardinality ≤ κ. We have to find a polynomial which extends f. Since μ^{κ} = μ we have κ < cf(μ), so there is a γ such that dom(f) ∪ ran(f) ⊆ L_{γ}^{n}. Let f = f_{γ,n}^{β}. Letting α := Γ(β, γ, n) ≥ γ we get a polynomial p_{α} with coefficients in L_{α+1} such that the induced polynomial function (in L_{α+1} as well as in L) extends f.

⌣ · ·^{}_{0.2}

References

[1] Marcel Erné and Dietmar Schweigert. Pre-fixed points of polynomial functions on lattices. Algebra Universalis, 31:298–300, 1994.

[2] Ervin Fried and Harry Lakser. Polynomial automorphism of lattices. Algebra Universalis, 27:371–384, 1990.

[3] G. Grätzer. General Lattice Theory. Series on Pure and Applied Mathematics. Academic Press, New York, 1978.

[4] H. Kaiser and R. Lidl. Erweiterungs- und Rédeipolynomvollständigkeit universaler Algebren. Acta Mathematica Scientiarum Hungaricae, 30:171–176, 1993.

[5] Hans Kaiser and Norbert Sauer. On order polynomially complete lattices. Algebra Universalis, 30:171–176, 1993.

[6] M. Kindermann. Über die Äquivalenz von Ordnungspolynomvollständigkeit und Toleranzeinfachheit endlicher Verbände. In Contributions to General Algebra, pages 145–149, 1978.

[7] Dietmar Schweigert. Über endliche, ordnungspolynomvollständige Verbände. Monatshefte für Mathematik, 78:68–76, 1974.

[8] R. Wille. Eine Charakterisierung endlicher, ordnungspolynomvollständiger Verbände. Archiv der Mathematik, 28:557–560, 1977.

Martin Goldstern Institut für Algebra und Diskrete Mathematik Technische Universität Wiedner Hauptstraße 8–10/118.2 A 1040 Wien

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