ON IRRESPONSIBLE HOMOMORPHISMS AND STRONG DUALITY

by

Brett Kelly
B.Sc., University of Northern British Columbia, 2014

THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
IN
MATHEMATICS

UNIVERSITY OF NORTHERN BRITISH COLUMBIA
July 2018

c Brett Kelly, 2018

Abstract

This thesis looks at algebras with positive primitively defined binary relations that are almost reflexive, anti-symmetric, and transitive and provides new machinery for determining when these
algebras are not strongly dualizable.

i

TABLE OF CONTENTS

Abstract

i

Table of Contents

ii

List of Figures

iv

List of Tables

v

Dedication

vi

Acknowledgement

vii

1

The Context of this Work

1

2

Notation and Background
2.1
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Product Sets and Operations . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Algebras and Subalgebras, Unary Algebras and Products . . . . . . . . . .
2.1.3 Homomorphisms and the Rows of an Algebra . . . . . . . . . . . . . . . .
2.1.4 Lattices and Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Topology, Quasi Varieties, and Topological Quasi Varieties . . . . . . . . . . . .
2.3
Natural, Full, and Strong Dualities . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Natural Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Full Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Strong Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Theorems About Strong Duality . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
4
5
7
8
9
12
13
14
15
18

3

Height, Irresponsibility, and Strong Duality
3.1
Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
PP-Formulæ and Irresponsibility . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Irresponsibility and Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
23
25

4

Height of Algebras and Irresponsibility
30
4.1
On Irresponsible and Responsible Fragments . . . . . . . . . . . . . . . . . . . . 30
4.2
The New Dense Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
ii

5

Unary Algebras and Irresponsibility
5.1
Irresponsible Fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Algebras with Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3
Escalator Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
One More Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Conclusion
44
6.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography

38
38
39
40
41

47

iii

LIST OF FIGURES

2.1
2.2
2.3
2.4
2.5

The dual of A is in ISc P+ (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The dual of X is in ISP(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A commuting diagram for the rank setup . . . . . . . . . . . . . . . . . . . . . .
h0 lifts to C/Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A commuting diagram for Enough Algebraic Operations . . . . . . . . . . . . . .

3.1
3.2
3.3

h lifts through Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Elements cri and crj with i < j. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Two example algebras Cν and Cµ from Lemma 3.3.3 . . . . . . . . . . . . . . . 29

5.1
5.2

The escalator algebra M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
The associated diagram for M5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 42

iv

13
13
16
16
17

LIST OF TABLES

2.1

M2.1.1 = h{0, 1, 2, 3}, f1 , f2 , f3 i . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

M3.2.1 = h{0, 1, 2, 3}, f0 , f1 , f2 , f3 i . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1

M5.4 = h{0, 1, 2, 3, 4, 5}, f , gi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1

A list of important symbols used in this thesis . . . . . . . . . . . . . . . . . . . 46

v

6

Dedication

This thesis is dedicated to my parents, Lynn and Jennifer Kelly, without whom I would never
have had the opportunity to attend the University of Northern British Columbia nor the support to
pursue mathematics.

vi

Acknowledgement

I would like to thank my original supervisor Jennifer Hyndman. I would also like to thank Matt
Reid for stepping in to take on supervisory duties when Dr Hyndman was away, a massive undertaking which required him to step outside of his usual area of research in physics. I would like to
thank David Casperson for his valuable insight into the research herein, and his incredibly helpful
discussions. I thank David Casperson and Brian Schaan for their help with LATEX. I appreciate my
friends, and my family Lynn, Jennifer, Robyn, and Jade Kelly for their support throughout.

vii

Chapter 1

The Context of this Work

In this document we look at algebras which have a positive primitively defined binary relation
that is almost reflexive, transitive, and antisymmetric. We choose these algebras with pp-formulae
partially as it has been shown that a particular kind of pp-formula prevents algebras from having
enough algebraic operations [7] but also because we can further develop the work in [1] to prove
that irresponsibility with respect to  forces an algebra to not be strongly dualizable by any alter
ego. The main result in this thesis, Theorem 4.2.1, provides a refinement of the main theorem in
[1] to a much more usable form by eliminating the most complicated condition.
Dualizable algebras allow us to consider problems in the language of some alter ego rather than
that of the original algebra and this process tends to reduce the complexity of our original problem.
If an algebra is strongly dualizable then this principle applies to the alter ego as well, that is we
may view problems of the topological alter ego in the language of the original algebra rather than
in terms of topology. Early examples of natural duality are documented in the work of Priestley
(of distributive lattices), Stone (of Boolean algebras), and Pontryagin (of Abelian groups). This is
similar in nature to Galois Theory, wherein a lattice of field extensions is dualized by a lattice of
1

automorphism groups which can be used to gain insight into the nature of the field generating the
original lattice. Of note, the Galois dual allows, for example, to prove the insolubility by radicals
of 5th degree and higher polynomials. Thus it is of great interest which algebras are dualizable
and which of these algebras are strongly dualizable and developing simpler tools to determine the
structures which cannot be (strongly) dualized is important.
The authors of [8] use an escalator algebra to provide an example of an algebra which is dualizable but not fully dualizable (and thus not strongly dualizable), an example which we revisit in
Chapter 5. In addition, the author of [12] found an example of an algebra which is fully dualizable
but not strongly dualizable. These two examples motivate the desire to determine conditions to
prove strong duality or, as in this document, simple tools for determining when an algebra is not
strongly dualizable.
On strong dualities, we briefly discuss rank in Chapter 2. Rank was one of the first tools used
to prove an algebra is strongly dualizable. The authors of [11] developed an improved tool, height,
for determining strong duality which we explore in Chapter 3. We then use height in Chapter 4
to develop our new mechanisms. In Chapter 5 we will focus our view on unary algebras and provide two examples where we can guarantee the existence of an irresponsible homomorphism and
immediately apply Theorem 4.2.1 to prove that these algebras are, in fact, not strongly dualizable.

2

Chapter 2

Notation and Background

This chapter lays out the necessary background in algebra, topology, and duality theory used in
this thesis. Most of the material on algebra comes from [3], the material on topology and duality
theory are from [5] and where appropriate the notation is updated to that of [11].

2.1

Algebra

We begin with a review of products of sets, projection maps, and operations. From here we will
lead into the definition of algebras, unary algebras, product algebras, and the rows of an algebra.

3

2.1.1

Product Sets and Operations
n

Given a collection of non-empty sets Ak for 1 ≤ k ≤ n we define the product, ∏ Ak , to be the set
k=1
n

∏ Ak = {(a1, a2, . . . , an) | ak ∈ Ak for 1 ≤ k ≤ n}

k=1
n

The elements of ∏ Ak are called n-tuples, and occasionally we choose to write a1 a2 . . . an instead
k=1

of (a1 , a2 , . . . , an ). For each i, 1 ≤ i ≤ n we define the projection map
n

πi : ∏ Ak → Ai
k=1

to be
πi (a1 , a2 , . . . , an ) = ai
n

for any n-tuple in ∏ Ak . In the case that Ai = A for each 1 ≤ i ≤ n then we simply write An instead
n

k=1

of ∏ Ai , and we define A0 to be the canonical one-element set. A map f : An → A is called an
i=1

n-ary operation on A, and the integer n is called the arity of f . If the arity of f is 0 then it is
called nullary and if the arity is 1 then f is called unary. If f is a unary operation that sends each
element of A to a particular element of the range then f is called a constant map, for example
f : A → A defined by f (x) = 0 for each x in A is a constant map. The map id : A → A defined by
id(x) = x for each x in A is called the identity map. Finally, if h : An → A is any n-ary operation
and a is an element of (An )I , for some indexing set I, we define h(a) to be the element of (An )I
such that for each i ∈ I we have
h(a)(i) = h(a(i)).

4

2.1.2

Algebras and Subalgebras, Unary Algebras and Products

A collection of operation symbols, F , is called a language or type if for each operation symbol
f in F there is a nonnegative integer n associated with it called the arity of f . An algebra, A,
is a pair hA, F i where A is a nonempty set, called the universe of A and each operation symbol
f ∈ F of arity n has an associated n-ary operation f A : An → A. We sometimes use F to denote
the set of all operations on the particular universe A, and so sometimes we will write hA, Fi instead
of hA, F i for the algebra A. The notation f A is used to make it clear that the operation is on the
algebra A. When there is no risk of ambiguity we will drop the superscript and simply write f . A
term function, t A , is any finite composition of the operations in the language of A using at most
finitely many variables x1 , x2 , . . . , xn . Again, if there is no risk of ambiguity we drop the superscript
and simply write t. Given an algebra A = hA, Fi and a set, G, of operations on A, we define the
algebra A0 = hA, F ∪ Gi to be an extension of A and the algebra A is called a reduct of A0 .
Let A = hA, F1 i and B = hB, F2 i be two algebras. They are said to be of the same type if
F1 and F2 are the same. Furthermore, if B is contained in A and for each operation on B we
have f B = f A B then B is called a subalgebra of A, denoted B ≤ A. A subuniverse of A is
any (possibly empty) subset of A that is closed under the operations of the algebra A. Therefore,
whenever B is a subalgebra of A, B is also a subuniverse of A.
Note that operations on products of algebras of the same type are defined coordinate-wise as
above. So if {Ai }i∈I is a collection of algebras of the same type, each f Ai : (Ai )n → Ai is an n-ary
operation, and A = ∏i∈I Ai then the n-ary operation f A : An → A is defined by

f A (a)(i) = f Ai (ai ).

Furthermore, operations on subalgebras are simply restrictions of the operations on the original
5

algebra. This means that any formula in the language F can be interpreted in any algebra of the
same type, we will revisit this in Chapter 3 when we discuss positive primitive formulae in detail.
When each operation in F has arity 1, we call the algebra a unary algebra, and when the range
of each operation symbol is contained in {0, 1} we call the algebra {0, 1}-valued. If |A| is finite,
then the algebra A is called finite. Furthermore, if |F | is finite we say the algebra A is of finite
type.
Example 2.1.1. Let M2.1.1 = h{0, 1, 2, 3}, f1 , f2 , f3 i be the algebra defined in Table 2.1. Since each
operation is unary the algebra is a unary algebra. In addition, each operation maps each element
of {0, 1, 2, 3} to either 0 or 1, so the algebra is {0, 1}-valued. Thus, M2.1.1 is a {0, 1}-valued unary
algebra. Also, since f1 ( f1 (a)) = 0 for each a ∈ M2.1.1 this algebra has the constant operation
g(x) = 0 as a term function. Furthermore, we notice that the sets {0, 1}, {0, 1, 2}, and {0, 1, 3} are

0
1
2
3

f1
0
0
1
0

f2
0
1
1
1

f3
0
1
0
0

Table 2.1: M2.1.1 = h{0, 1, 2, 3}, f1 , f2 , f3 i

closed under the operations of M2.1.1 and so they all form subalgebras.
n

Given a collection of algebras of type F , Ai for 1 ≤ i ≤ n then the product A = ∏ Ai is also an
i=1

algebra of type F , called the product algebra of the Ai . For each n-ary operation symbol f ∈ F
and a1 , a2 , . . . , an in A we define

f A (a1 , a2 , . . . , an )(i) = f Ai (a1 (i), a2 (i), . . . , an (i))

6

2.1.3

Homomorphisms and the Rows of an Algebra

Let A and B be two algebras of the same type, F . A map h : A → B is a homomorphism if for
each n-ary operation symbol in F we have

h( f A (a1 , a2 , . . . , an )) = f B (h(a1 ), h(a2 ), . . . , h(an )).

This property in particular is called operation preserving. If in addition h is bijective, then we
call it an isomorphism. If there is an isomorphism between the algebras A and B we say they are
isomorphic and denote this A ∼
= B. If there is a homomorphism h : A → B that is onto, then we
call B a homomorphic image of A. A one-to-one homomorphism is called an embedding.
The authors of [4] define the rows of an algebra as follows. Let M = hM, Fi be a finite unary
algebra of finite type. Let F 0 = { f1 , f2 , . . . , fν } be a fixed enumeration of the non-constant, non
identity operations in F. For each a ∈ M we define the row at a, denoted row(a), to be the ν-tuple

row(a) = ( f1 (a), f2 (a), . . . , fν (a)).

We then define the ν-ary relation, Rows(M) as

Rows(M) = {row(a) | a ∈ M}
= {( f1 (a), f2 (a), . . . , fν (a)) | a ∈ M}.

If M is a finite unary algebra of finite type and there is an element 0 ∈ M such that row(0) = (0, 0, . . . , 0)
then M is a unary algebra with 0. If there are elements a, b ∈ M such that a 6= b and row(a) = row(b)
then we say the algebra M has a repeated row, or sometimes that M has non-unique rows. If M
does not have a repeated row, then we say the algebra has unique rows. If M has unique rows then
7

we say that the rows of M are uniquely witnessed.

2.1.4

Lattices and Semilattices

The following definitions of lattices and semilattices come from [3].
A binary operation ≤ defined on a set A is a partial order if for every a, b, and c in A the
following are true

1. a ≤ a (≤ is reflexive),
2. a ≤ b and b ≤ a implies a = a (≤ is antisymmetric), and
3. a ≤ b and b ≤ c implies a ≤ c (≤ is transitive).

A set with a partial order is called a partially ordered set or simply poset and we often denote this
hA, ≤i. If for every pair of elements a, b ∈ A either a ≤ b or b ≤ a then ≤ is called a total order.
An algebra L = hL, ∧i is a semilattice provided ∧ : L2 → L satisfies the following properties

1. x ∧ y ≈ y ∧ x (∧ is commutative),
2. (x ∧ y) ∧ z ≈ x ∧ (y ∧ z) (∧ is associative), and
3. x ∧ x ≈ x (∧ is idempotent).

The algebra L = hL, ∧, ∨i, with ∧, ∨ both binary is a lattice if it satisfies the following,

1. hL, ∧i is a semilattice,

8

2. hL, ∨i is a semilattice, and
3. x ∧ (x ∨ y) ≈ x and x ∨ (x ∧ y) ≈ x (∧ absorbs ∨ and ∨ absorbs ∧).

The operations ∧ and ∨ are called meet and join respectively and a semilattice hL, ∧i or hL, ∨i is
often called meet semilattice and join semilattice respectively.
If L is a lattice then we can define a partial order on L by a ≤ b whenever a = a ∧ b.
If A is a unary algebra we say that A has a meet homomorphism if there is a homomorphism
∧ : A2 → A such that hA, ∧i is a semilattice.

2.2

Topology, Quasi Varieties, and Topological Quasi Varieties

Our discussion of topology will be limited to the absolute basics since we will never be doing any
topology explicitly. Once we have what we need we will discuss the categories of (algebraic) quasi
varieties and topological quasi varieties.
Given a set X, a collection of subsets of X, T , is a topology provided:

1. 0/ and X are in T ,
2. If A and B are in T , then A ∩ B ∈ T , and
3. If Ai ∈ T for i in some indexing set I, then

i∈I Ai ∈ T .

S

If the only sets in the topology are the empty set and X then it is called trivial. If instead every
subset of X is in the topology then it is called discrete.

9

Let M be a finite algebra. A homomorphism g : Mn → M for n ∈ ω is called a (total) algebraic
operation. If A ≤ Mn for n ∈ ω, a homomorphism h : A → M is called a partial algebraic
operation. Lastly, a subset r ⊆ M n that forms a subuniverse of Mn for n ∈ ω is called an algebraic
relation. The integer n is called the arity of the relation.
Let G, H, R be sets of algebraic operations, partial operations, and relations on M respectively
and let T be the discrete topology on M. The topological structure M = hM, G, H, R, T i is called
an alter ego of M. The set G ∪ H ∪ R is called the type of M, and if |G ∪ H ∪ R| is finite then M
is said to be of finite type. The (algebraic) quasi variety generated by M is the class of all isomorphic copies of subalgebras of products of the finite algebra M, and is denoted ISP(M). Note
that the operators ISP(M) is closed under S and P, that is S(ISP(M)) = ISP(M) = P(ISP(M)).
The topological quasi variety generated by M is the class of all isomorphic copies of closed substructures of positive powers of the alter ego M, and is denoted ISc P+ (M). Similarly, ISc P+ (M)
is also closed under the operators Sc and P+
In our investigation of algebras we reviewed how isomorphism, subalgebras, and products related to the algebraic structure of a given algebra. But how do isomorphic copies, closed substructures, and positive powers work with the topological structure of the alter ego? We will answer this
question from the “inside out,” first with positive products, then closed substructures, and finally
isomorphic copies.
Suppose M is an alter ego of the finite algebra M. Let g : M n → M be an algebraic operation in G, for some n ∈ ω and let S be a non-empty set. We define the algebraic operation
S

gM : (M S )n → M S by

S

gM (m1 , m2 , . . . , mn )(s) = g(m1 (s), m2 (s), . . . , mn (s))

10

for all m1 , m2 , . . . , mn ∈ M S and s ∈ S.
Let h : dom(h) → M be an n-ary partial algebraic operation in H, with dom(h) ⊆ M n for some
S

n ∈ ω. The domain of the partial algebraic operation hM is defined as

 S
dom hM = {(a1 , a2 , . . . , an ) ∈ (M S )n : (a1 (s), a2 (s), . . . , an (s)) ∈ dom(h) ∀s ∈ S}

 S
S
and the partial algebraic operation hM : dom hM → M S is defined by

S

hM (m1 , m2 , . . . , mn )(s) = h(m1 (s), m2 (s), . . . , mn (s))

 S
for all (m1 , m2 , . . . , mn ) ∈ dom hM and s ∈ S.
S

Given an n-ary relation r in R, for n ∈ ω, we define the relation rM in the same manner:

S

rM = {(a1 , a2 , . . . , an ) ∈ (M S )n : (a1 (s), a2 (s), . . . , an (s)) ∈ r, ∀s ∈ S}.

Finally, the topology on MS is derived from the topology on M by the standard product topology.
This topology is determined by the clopen subbasis

Us,m = {a ∈ M S : a(s) = m}.

Notice that the topology on MS is always compact and Hausdorff, and if S is finite then it is discrete,
definitions of which can be found in [10].
Each closed substructure Y of X is closed in the topology of X and under the algebraic total and partial operations in G ∪ H, that is for g an n-ary total operation in G then whenever
(a1 , a2 , . . . , an ) ∈ Y we have g(a1 , a2 , . . . , an ) ∈ Y . If h is an n-ary partial operation in H then

11

whenever (a1 , a2 , . . . , an ) ∈ Y and (a1 , a2 , . . . , an ) ∈ dom(h) we have h(a1 , a2 , . . . , an ) ∈ Y .
Given two structures X and Y of the same type, the map α : X → Y is a morphism if α is
continuous, and for each n-ary operation in G

α(gX (x1 , x2 , . . . , xn )) = gY (α(x1 ), α(x2 ), . . . , α(xn )),



for each n-ary partial operation in H, whenever (x1 , x2 , . . . , xn ) ∈ dom hX we have (α(x1 ), α(x2 ), . . . ,


α(xn )) ∈ dom hY and

α(hX (x1 , x2 , . . . , xn )) = hY (α(x1 ), α(x2 ), . . . , α(xn )),

and for each n-ary relation r in R, if (x1 , x2 , . . . , xn ) ∈ rX then (α(x1 ), α(x2 ), . . . , α(xn )) ∈ rY . If
a bijective morphism exists between the structures X and Y we borrow from algebra and call
them isomorphic, and call the morphism an isomorphism. A one-to-one morphism is called an
embedding.

2.3

Natural, Full, and Strong Dualities

In this section we lay out the groundwork for developing dualities, full dualities, and strong dualities. While this thesis does not prove dualizability, it is a necessary condition for strong duality
and so is detailed here.

12

2.3.1

Natural Dualities

Let M be a finite algebra and M an alter ego of the algebra M. First we define the contravariant
functors D and E between the categories ISP(M) and ISc P+ (M). For an algebra A ∈ ISP(M) the
set hom(A, M) is the set of all homomorphisms with domain A and codomain M. We define the
dual of A in ISc P+ (M) by
D(A) = hom(A, M).
For a structure X ∈ ISc P+ (M) the set hom(X, M) is the set of all morphisms with domain X and
ISc P+ (M)

ISP(M)
A

D

D(A)

Figure 2.1: The dual of A is in ISc P+ (M)

codomain M. We define the dual of X in ISc P+ (M) by

E(X) = hom(X, M).

ISc P+ (M)

ISP(M)
E(X)

E

X

Figure 2.2: The dual of X is in ISP(M)

13

For each homomorphism ϕ : A → B, with A, B ∈ ISP(M) the morphism D(ϕ) : D(B) → D(A)
is defined by
D(ϕ)(x) = x ◦ ϕ
for each x ∈ hom(B, M). For each morphism ψ : X → Y, with X, Y ∈ ISc P+ (M) the homomorphism E(ψ) : E(Y) → E(X) is defined by

E(ψ)(α) = α ◦ ψ

for each α ∈ hom(Y, M).
Let A ∈ ISP(M), the evaluation map eA : A → ED(A) is given by

eA (a)(x) = x(a)

for a ∈ A and x ∈ hom(A, M). If A ∼
= ED(A) via the evaluation map then we say that M yields a
duality on A. If M yields a duality on every A ∈ ISP(M) then we say that M yields a duality on
ISP(M), or that M dualizes M. If there is an alter ego that yields a duality on M we say that M is
dualizable.

2.3.2

Full Dualities

There is also a natural embedding for the topological quasi variety, for X ∈ ISc P+ (M) the evaluation map εX : X → DE(X), given by

εX (x)(α) = α(x)

14

for x ∈ X and α ∈ hom(X, M). If M yields a duality on M and εX is an isomorphism for each
X ∈ ISc P+ (M) then we say M yields a full duality on ISP(M), or that M fully dualizes M. If
there is an alter ego that yields a full duality on M we say that M is fully dualizable.

2.3.3

Strong Dualities

For a pair of maps x : A → B and y : A → B, the equalizer of x and y, is given by

eq(x, y) = {a ∈ A : x(a) = y(a)}

A subset X of M S , for a non-empty set S is term closed if for each y ∈ M S \ X there are S-ary term
functions t1 ,t2 such that t1 and t2 agree on X but differ at y. That is, X is term closed if

X=

\

{eq(t1 ,t2 ) | t1 ,t2 S-ary and t1 X = t2 X }

If M yields a duality on M and each closed substructure of each non-zero power of M is term
closed then M yields a strong duality on M, or M is strongly dualized by M. If there is an alter
ego that yields a strong duality on M then we say that M is strongly dualizable.
The following is the definition of rank as it appears in [6]. Let M be a finite algebra, n a positive
integer, B a subalgebra of Mn , and h : B → M a homomorphism. We write B Vσ B0 to denote that
B0 is a subalgebra of Mn+k for some finite integer k, σ embeds B in B0 by repetition of some coordinates and B ∼
= B0 . Let h0 = σ −1 ◦ h be the natural extension of h to B0 . Let B0 ≤ C ≤ D ≤ Mn+k
and assume there exists a homomorphism h+ : D → M such that h0 = h+ B0 . When we refer to the
commuting diagram in Figure 2.3 we assume the above setup holds.
Let Y ⊆ hom(D, M), D/Y is defined as the algebra D/ {ker(g) | g ∈ Y } and C/Y is the algebra
T

15

B

Vσ

B0

≤

C

≤

D

h0
h+

h

M

Figure 2.3: A commuting diagram for the rank setup
C/ {ker(gC | g ∈ Y }. The set Y separates B0 if for every pair of elements a, b ∈ B0 there is a
T

homomorphism g ∈ Y such that gB0 (a) 6= gB0 (b). The homomorphism h0 lifts to C/Y if Y
separates B0 and there exists a homomorphism µ such that Figure 2.4 commutes.
B0

≤

C

h0

M

µ

C/Y

Figure 2.4: h0 lifts to C/Y

Given a homomorphism h : B → M, we define the rank of h, denoted rank(h), as follows:
rank(h) ≤ 0 if and only if h is a projection. rank(h) ≤ α if and only if there exists a finite N such
that for all nonnegative integers k, for all subalgebras D of Mn+k , and for all commuting diagrams
like Figure 2.3, where h0 lifts to D, there exists Y ⊆ hom(D, M) such that:

• |Y | ≤ N,
• h0 lifts to C/Y , and
• rank(gC ) < α for all g ∈ Y .

Further, rank(h) = α if rank(h) ≤ α and rank(h) 6< α. The rank of the algebra M, denoted
16

rank(M), is the least α such that rank(h) ≤ α for every homomorphism h ∈ hom(A, M) where
A ≤ Mn for some n ∈ ω. If rank(M) is finite then we say M has finite rank.
Now we present the definition of enough algebraic operations, as found in [9]. For each subset
Y of hom(Mn , M) the natural product homomorphism, uY : Mn → MY is the map

uY (a)(y) = y(a)

for each a ∈ M n and y ∈ Y . We say that M has enough algebraic operations if there is a
map f : ω → ω such that for all nonnegative integers n, algebras B ≤ A ≤ Mn , and all homomorphisms h : A → M there is a set Y ⊆ hom(Mn , M) with |Y | ≤ f (|B|) and a homomorphism
h0 : uY (A) → M such that h0 ◦ uY B = hB . That is, the diagram in Figure 2.5 commutes.
B

uY A

hB
M

Mn

A

h0

uY (A)

Figure 2.5: A commuting diagram for Enough Algebraic Operations

There is another concept, called height, which is related to rank and enough algebraic operations, but we will reserve our discussion of height to the next chapter of this document.

17

2.4

Theorems About Strong Duality

To prove a given algebra M is strongly dualizable we need to not only find an alter ego M that dualizes the algebra, but we need to prove that each closed substructure X ∈ ISc P+ (M) is actually term
closed. In order to prove that a dualizable algebra M is not strongly dualizable we need to show
that for any alter ego M that dualizes the algebra there is some closed subtructure X ∈ ISc P+ (M)
that is not term closed. In this section we detail some theorems that allow us to prove that an
algebra is strongly dualizable or that it is not strongly dualizable without needing to investigate
every possible alter ego. We first make a small definition and note its importance. Throughout this
section M denotes a finite algebra.
Let Hω be the set of all algebraic partial operations on M with finite arity. The strong bruteforce alter ego of M, defined in [11], is the structure

MΩ = hM, Hω , T i.

This alter ego is special in that if some alter ego strongly dualizes M, then so does MΩ . Since we
are only concerned with whether our algebra is strongly dualizable, not finding a useful structure
that strongly dualizes M, unless we specify the alter ego we will be working with respect to MΩ .
We defined rank in the previous section, the following theorem allows us to determine the
strong dualizability of M based on its rank.
Theorem 2.4.1. [13] Let M be a finite algebra. If M is dualizable and has finite rank, then M is
strongly dualizable.

With rank in hand we note the stronger, but easier to manage, concept of enough algebraic
operations and its significance.
18

Theorem 2.4.2. [9], [11] Let M be a finite algebra. If M has enough algebraic operations, then
M has rank at most 2.

Since enough algebraic operations is sufficient to show finite rank, we notice the following
Theorem 2.4.3. [9], [11] Let M be a finite algebra. If M is dualizable and has enough algebraic
operations, then M is strongly dualizable.

Rank and enough algebraic operations are useful for showing an algebra is strongly dualizable,
but if M does not have enough algebraic operations or does not have finite rank we cannot determine if the algebra is strongly dualizable. Fortunately, in [11] a condition, height, that is equivalent
to strong duality is developed.
Theorem 2.4.4. [11] Let M be a finite algebra. M is strongly dualizable if and only if it is dualizable and has a height.

In the following chapter we will discuss the concept of height, as well as present the main
findings from [1] that inspired the research involved in this thesis.

19

Chapter 3

Height, Irresponsibility, and Strong Duality

In this chapter we outline the concepts of height and irresponsibility that will form the backbone of
the proofs supplied in Chapter 4. The conclusion of this chapter will be the main theorem from [1]
that inspires the main results of this document. We utilize the fact that a dualizable algebra does
not have a height if and only if it is not strongly dualizable in order to prove particular algebras
are not strongly dualizable. Throughout the rest of this thesis we assume that M is a finite algebra,
and whenever we refer to the dual of an algebra in ISP(M) we take it with respect to the strong
brute-force alter ego. All of the definitions in this chapter are found in [1] and [7]

3.1

Height

Let A be an algebra in SP(M), the category of all subalgebras of products of the algebra M.
For a subalgebra C of A we use the notation C  A to denote that C is finite. For C  A a
homomorphism h : C → M is a D(A)-fragment if there is a homomorphism h+ : A → M such that
h = h+ C [1]. We denote the set of all D(A)-fragments by FragD(A) . Finally, if B is a subalgebra
20

of A, h : B → M and Y ⊆ hom(A, M) we say that h lifts through Y if there is a homomorphism
p : uY (A) → M such that the diagram in Figure 3.1 commutes [1].
B

A
uY A

h
M

p

uY (A)

MY

Figure 3.1: h lifts through Y

In [11] the authors describe the height of any homomorphism by beginning with a strong duality. Taking X as the smallest closed substructure of hom(A, M) that contains the projections we
have by strong duality that X is also term closed. Since X is term closed and contains the projections we have that X = hom(A, M) [11]. So by beginning with the projections and alternately
closing under the algebraic operations and closing under topology we will eventually reach all of
D(A). The number of steps necessary to construct the homomorphism h is the height of h, and the
total number of steps to construct all of D(A) is the height of the algebra A. Instead of using the
process of [11] we use a simpler definition found in [1].
The following is the definition of height as it appears in [1]. Let A be a subalgebra of MI
for some set I and let h : B → M be a D(A)-fragment. We define the height of h in FragD(A) by
transfinite induction as follows. The homomorphism h has height at most 0 in FragD(A) if it is a
coordinate projection. For every ordinal α > 0, we say that h has height at most α in FragD(A)
if there exists a nonnegative integer r such that for all algebras C with B ≤ C  A there exists an
ordinal β < α and a collection of D(A)-fragments Y ⊆ hom(C, M) such that

1. |Y | ≤ r,
21

2. every homomorphism in Y has height at most β in FragD(A) , and
3. h lifts through Y .

The height of h in FragD(A) is the least ordinal α such that h has height at most α in FragD(A) . If
no such α exists then we say that h does not have a height in FragD(A) .
We say that M does not have a height if there is an algebra A ∈ SP(M) and a D(A)-fragment
h that does not have a height in FragD(A) . Next we define when a collection of D(A)-fragments is
dense in FragD(A) , a condition that aids in determining when M does not have a height. We note
that if a set is not dense in FragD(A) then the corresponding algebra may still have a height.
Let A ∈ SP(M) and h : B → M with B  A. A collection U of D(A)-fragments is dense at
h if for all positive integers r there is an algebra C with B ≤ C  A such that for all nonempty
collections of D(A)-fragments Y ⊆ hom(C, M) for which we have that |Y | ≤ r and h lifts through
Y implies that Y ∩U is nonempty. A collection U of D(A)-fragments is dense in FragD(A) if it is
dense at each of its members [1]. Note:
Theorem 3.1.1. [1] Let M be a finite algebra and A ∈ SP(M). If U is dense in FragD(A) and does
not contain any projections or constant homomorphisms, then h does not have a height in FragD(A)
for every h ∈ U.

And so an immediate consequence of Theorem 3.1.1 is:
Corollary 3.1.2. [1] Let M be a finite algebra and A ∈ SP(M). If U is a nonempty, dense subset of
FragD(A) that does not contain any projections or constant homomorphisms, then M is not strongly
dualizable.

22

3.2

PP-Formulæ and Irresponsibility

A primitive positive formula, abbreviated pp-formula, in the language of an algebra M is an
existentially quantified conjunction of equations in the language of M [7]. For a unary algebra
such an equation is of the form f (x) ≈ g(y) for term operations f and g, either of which may be
the identity operation and variables x and y which need not be distinct. For instance, a pp-formula
may take the form
Φ : ∃ w1 , w2 [ f (x) ≈ w2 & y ≈ g(w1 ) & x ≈ h(x)].
Here w1 and w2 are existentially quantified variables, x and y are free variables, and f , g, and h are
unary term operations. We denote this pp-formula as Φ(x, y), or simply Φ. Note that we exclude
any existentially quantified variables from the argument. In general a pp-formula can have any
number of existentially quantified variables.
Given a pp-formula Φ(x1 , x2 , . . . , xn ) with existentially quantified variables w1 , w2 , . . . , wm and
a1 , a2 , . . . , an in M, we say that Φ(a1 , a2 , . . . , an ) holds in M, or that M satisfies Φ(a1 , a2 , . . . , an )
denoted M |= Φ(a1 , a2 , . . . , an ), if substituting each xi with the corresponding ai in Φ results in a
true statement [7]. The value(s) of the tuples (w1 , w2 , . . . , wm ) such that Φ(a1 , a2 , . . . , an ) holds in
M are called witnesses of Φ(a1 , a2 , . . . , an ). For an algebra A ∈ SP(M) and elements a1 , a2 , . . . , an
in A, A |= Φ(a1 , a2 , . . . , an ) if there are witnesses (w1 , w2 , . . . , wm ) in A such that Φ(a1 , a2 , . . . , an )
is true in A.
Example 3.2.1. Let M3.2.1 = h{0, 1, 2, 3}, f0 , f1 , f2 , f3 i be the algebra defined in Table 3.1. Define
the pp-formula Φ(x, y) by
Φ : ∃ w[x ≈ f2 (w) & y ≈ f3 (w)].
We can see that M3.2.1 satisfies Φ(0, 0), Φ(0, 1), and Φ(1, 1) witnessed by 0 and 2, 1, and 3,
respectively. Notice that we can define the usual ≤ relation on {0, 1} by a ≤ b if M |= Φ(a, b).
23

0
1
2
3

f0
0
0
0
0

f1
0
0
1
0

f2
0
0
0
1

f3
0
1
0
1

Table 3.1: M3.2.1 = h{0, 1, 2, 3}, f0 , f1 , f2 , f3 i
Given a pp-formula Φ in two free variables, we can define a binary relation  on M like we
did in Example 3.2.1 by

a  b if and only if a, b ∈ M and M |= Φ(a, b).

In general, we can define the relation  for any algebra A in the quasi variety ISP(M) in a similar
way. So we say,
a  b in A

if and only if

a, b ∈ A and A |= Φ(a, b).

Recall that by the definition of operations on product algebras and subalgebras the formula Φ
described in the language of M can in fact be interpreted in the language of any algebra A in
ISP(M) by using the appropriate term function, f A . Finally we note that for a, b ∈ MI we have
that a  b in MI if and only if a(i)  b(i) in M for every i ∈ I.
Note that due to the definition of a pp-formula, and ai  bi in M for each i in some indexing
set I then the product of witnesses w j = (wi, j )i∈I is the jth witness of a = (ai )i∈I  (bi )i∈I = b.
Furthermore, if h : MI → M is a homomorphism, and a  b in MI , then h(a)  h(b) since if wi, j is
the jth witness for a(i)  b(i) in M, then w j = (wi, j )i∈I is the jth witness for a  b in MI . Since h
is a homomorphism it preserves the operations in the type of M, and so h(w j ) is the jth witness for
h(a)  h(b). However, if A is a subalgebra of MI and a, b ∈ A with a  b in MI and g : A → M,
then g(a)  g(b) in M need not be true since the witnesses w j of a  b in MI may not be an element

24

of A.
With this in mind, for A a subalgebra of MI and h : A → M we say that h is responsible with
respect to  if for all a, b ∈ A whenever a  b in MI we have h(a)  h(b) in M. Otherwise,
we say that h is irresponsible with respect to , or simply that h is irresponsible. If we are
concerned with a particular pair that demonstrates that h is irresponsible, then we may say that h is
irresponsible with respect to a  b.

3.3

Irresponsibility and Height

A given binary relation is called almost reflexive if whenever (a, b) is in the relation so are (a, a)
and (b, b) [1]. Let M be a finite algebra with a pp-definable, transitive, antisymmetric, almost
reflexive binary relation  and assume there is an irresponsible homomorphism h0 : B0 → M with
B0 ≤ Mn and a0 , b0 distinct elements of B0 such that h0 is irresponsible with respect to a0  b0 .
We can take B0 = SgMn ({a0 , b0 }), the subalgebra of Mn generated by the elements a0 , and b0 . The
following construction builds toward a result that guarantees a set that is dense in FragD(A) . For
a ∈ M, the notation a will denote the tuple in M J with value a at each coordinate.
Construction 3.3.1. [1] Let I = [0, 1) ∪ {1, . . . , n}. Let B ≤ MI = M[0,1) × Mn be an isomorphic
copy of B0 obtained by the following. Since a0  b0 in Mn and a0 6= b0 , we have that for all j,
1 ≤ j ≤ n a0 ( j)  b0 ( j), and for some ı, a0 (ı) 6= b0 (ı). Define the embedding σ : Mn → MI by



σ (c) = c(ı), c .

Fix a = σ (a0 ) and b = σ (b0 ), and define B = σ (B0 ). Note that σ : B0 → B is an isomorphism.
Finally, let h = h0 ◦ σ −1 B . We call h the associate of h0 .
25

The next construction allows us to build the desired algebra A in SP(M) to form the dense in
FragD(A) set.
Construction 3.3.2. [1] Fix m0 an even integer with m0 > |M|. Set k0 = 0 and for each integer
r ≥ 1, set kr = rm0 + 2. Note that each kr is even. For integers r and i, r ≥ 0 and 0 ≤ i ≤ 1 + kr set
i
and define cri ∈ M I as:
qri = 1+k
r

cr0 = a





a(x), for x ∈ [0, 1 − qri ),




r
ci (x) = b(x), for x ∈ [1 − qr , 1),
i







a(x), for x ≥ 1,

for 1 ≤ i ≤ kr , and

cr1+kr = b.

Note that, for 1 ≤ i ≤ kr the element cri has value a0 (ı) on [0, 1 − qri ) and value b0 (ı) on [1 − qri , 1)
as in the following figure.
a(x)

[
0
[
0

a(x)

|
1 − qrj

|
1 − qri
b(x)

b(x)

)
1

a(1)
×
1

a(2)
×
2

...

a(n)
×
n

)
1

a(1)
×
1

a(2)
×
2

...

a(n)
×
n

Figure 3.2: Elements cri and crj with i < j.

As we can see in the figure, it would appear that cri  cri+1 in MI for each i with 0 ≤ i ≤ kr .
We wish to build an algebra A so that cri  crj is witnessed in A for particular choices of i and j,
namely when j − i is even. To do so we construct our choices of witnesses.

26

Let Γ be an indexing set for the witnesses of the pp-formula that defines  that is, the ppformula Φ defining  has |Γ| existentially quantified variables and each γ ∈ Γ corresponds to one
of these variables. For γ ∈ Γ let γ wa0 ,a0 ,γ wa0 ,b0 , and γ wb0 ,b0 be a fixed γ th witness for a0  a0 , a0  b0 ,
and b0  b0 in Mn respectively. Let γ wa,a = σ (γ wa0 ,a0 ),γ wa,b = σ (γ wa0 ,b0 ), and γ wb,b = σ (γ wb0 ,b0 ).
For γ ∈ Γ and 0 ≤ i < j ≤ kr with j − i even, define γ wri, j ∈ M I by

γ

wri, j (x) =




γ w (x), for x ∈ [0, 1 − qr ),


a,a
j







γ wa,b (x), for x ∈ [1 − qrj , 1 − qri ),


γ w (x), for x ∈ [1 − qr , 1),


b,b

i






γ wa,a (x), for x ≥ 1,

and for i odd, with 0 < i < kr , we define γ wi,1+kr ∈ M I by

γ

wri,1+kr (x) =




γ w (x), for x ∈ [0, 1 − qr ),


a,b
i





γ w (x), for x ∈ [1 − qr , 1),
b,b
i







γ wa,b (x), for x ≥ 1.

Notice that for 0 ≤ i < j ≤ 1 + kr and j − i even, the element γ wri, j witnesses cri  crj in MI .
For r ≥ 0, let

Cr0 = {cri | 0 ≤ i ≤ 1 + kr },
Wr = {γ wri, j | γ ∈ Γ, 0 ≤ i < j ≤ 1 + kr with j − i even},
Cr = SgMI (Cr0 ∪Wr ),
[

A = SgMI
Cr ,
r≥0

27

completing the construction.

Now, C0 = B ≤ Cr , since W0 is empty. Also, cr0  cr1  · · ·  cr1+kr in MI and by choice of
witnesses cri  crj in Cr when i < j and j − i is even.
Let r > 0, and let Pr be the set of irresponsible D(A)-fragments with domain Cr . We define the
set Uh to be Uh = {h} ∪

S

r>0 Pr . Now we present the main theorem of [1].

Theorem 3.3.1. [1] Let M be a finite algebra. Assume there is a binary relation  on M that is
pp-definable, antisymmetric, transitive, and almost reflexive. Further assume that

1. For some B0 ≤ Mn , there exists a homomorphism h0 : B0 → M and there exist distinct elements a0 , b0 in B0 with a0  b0 in Mn , such that h0 (a0 ) 6 h0 (b0 ), and such that B0 = SgMn ({a0 , b0 }).
2. The associate h : B → M of h0 is a D(A)-fragment, where A, B, and Cr with r ≥ 0 are defined
as in Construction 3.3.1 and Construction 3.3.2.
3. For all r > 0 and u, v ∈ Cr with u  v in MI such that there is a D(A)-fragment g : Cr → M
with g(u) 6 g(v), there exist integers i, j with 0 ≤ i < j ≤ 1 + kr such that u = cri and v = crj .

Then the set Uh is dense in FragD(A) , and therefore M is not strongly dualizable.

There are two other results that will find some use in the following chapters, they are
Lemma 3.3.2. [1] Fix r1 , r2 > 0. If cri11 ∈ Cr0 1 and cri22 ∈ Cr0 2 with cri11 = cri22 , then i2 − i1 is even.
If there exists a term τ such that for all γ ∈ Γ, r ≥ 1, and i and l with 0 ≤ i < l ≤ 1 + kr and l − i
even, we have τ(γ wri,l ) ∈ {cri , crl }, then, whenever γ wri11, j1 and γ wri22, j2 are defined and equal, i2 − i1
is even.

28

and
Lemma 3.3.3. [1] Fix a non-negative integer ν and a positive integer s. There exist integers µ and
t with t odd, t > s, and µ > ν such that Cν ≤ Cµ and such that for all integers i with 0 ≤ i ≤ 1 + kν
µ

we have cνi = cti .

Lemma 3.3.3 allows us to create superalgebras that “fill in” the gaps between the cri s, to guarantee density at each irresponsible homomorphism with domain Cr .
Cν :

Cµ :

...

cν0

cν1

a

cν1 = c3·1

µ

c0

µ

µ

c1

µ

c2

µ

c3

cνkν

cνkν +1
µ

cνkν = c3·kν
µ

c4

µ

c5

b

. . . ckµ −2 ckµ −1 ckµ ckµ +1
µ
µ
µ
µ

Figure 3.3: Two example algebras Cν and Cµ from Lemma 3.3.3

29

Chapter 4

Height of Algebras and Irresponsibility

In this chapter we construct a different dense-in-FragD(A) set using tools outlined in the previous
chapter. While this construction applies to all algebras, regardless of type, in the next chapter
we apply the result to unary algebras. Throughout this chapter, we assume that M is an algebra
with a positive primitively defined binary relation,  that is almost reflexive, anti-symmetric, and
transitive.

4.1

On Irresponsible and Responsible Fragments

Let A ≤ MI . Recall that h : A → M is irresponsible if there exists a, b ∈ A such that a  b in MI
but h(a) 6 h(b) in M. Otherwise, the homomorphism is responsible. Thus, if a given homomorphism h : A → M extends to MI , then it must be responsible. Note that projections and constant
homomorphisms are responsible.

30

Given a set X with a binary relation  a subset C of X is called convex if,

a  c & c  b implies c ∈ C for all a, b ∈ C and c ∈ X

The following Lemma from [1] provides the maximum number of convex sets the kernel of uY
can partition an ordered set into, provided each y in Y is responsible.
Lemma 4.1.1. [1] Let M be a finite algebra and let  be an antisymmetric and transitive ppdefinable binary relation on M. Let C  A ≤ MI with d1 , . . . , dk ∈ C for some k, and assume
that
d1  d2  · · ·  dk in MI
Let Y ⊆ hom(C, M) be a finite collection of responsible D(A)-fragments. Then the kernel of uY
partitions {d1 , . . . , dk } into at most |Y |(|M| − 1) + 1 convex sets with respect to  on MI .

Lemma 4.1.2 is a subtle modification to this result. In Lemma 4.1.2 we take Y to be a collection
of D(A)-fragments that only need to be responsible with respect to di  d j for i < j rather than
fragments which are simply responsible.
Lemma 4.1.2. Let M be a finite algebra and let  be an antisymmetric and transitive pp-defined
binary relation on M. Let C  A ≤ MI with d1 , . . . , dk ∈ C for some k, and

d1  d2  · · ·  dk in MI

Let Y ⊆ hom(C, M) be a collection of D(A)-fragments responsible with respect to di  d j for
1 ≤ i < j ≤ k. Then the kernel of uY partitions {d1 , . . . , dk } into at most |Y |(|M| − 1) + 1 convex
sets with respect to  on MI .

31

Proof. Since each y ∈ Y is responsible with respect to d1  · · ·  dk , we have that y(di )  y(d j ) for
each i < j. Since  is antisymmetric and transitive on M it is antisymmetric and transitive on MI .
If y(di ) = y(d j ), then y(di )  y(dt )  y(d j ) = y(di ) for each i ≤ t ≤ j, and so y(di ) = y(dt ) = y(d j )
for each i ≤ t ≤ j. So each y partitions {d1 , . . . , dk } into at most |M| convex sets. Now, if uY (di ) 6=
uY (d j ) then there is some y such that y(di ) 6= y(d j ). There are at most |M| − 1 values i such that
y(di ) 6= y(di+1 ), so there are at most |Y |(|M| − 1) values i such that uY (di ) 6= uY (di+1 ). Therefore
the kernel of uY partitions {d1 , . . . , dk } into at most |Y |(|M| − 1) + 1 convex sets with respect to
.

Using Constructions 3.3.1, 3.3.2, and Lemma 4.1.2, we prove that irresponsible D(A)-fragments
have no height, and therefore certain algebras are not strongly dualizable. This result sets us up to
provide our alternative dense-in-FragD(A) set.

4.2

The New Dense Set

We utilize the same constructions provided in [1], as presented in Chapter 3. We restate those
constructions here.
Construction 3.3.1. [1] Suppose there is a pair of elements a0 , b0 ∈ Mn such that there is a
homomorphism h0 : B0 → M such that a0  b0 in Mn but h0 (a0 ) 6= h0 (b0 ), where B0 = SgMn ({a0 , b0 }).
Let I = [0, 1) ∪ {1, . . . , n}. Let B ≤ MI = M[0,1) × Mn be an isomorphic copy of B0 obtained by
the following. Since a0  b0 in Mn and a0 6= b0 , we have that for all j, 1 ≤ j ≤ n a0 ( j)  b0 ( j), and
for some ı, a0 (ı) 6= b0 (ı). Define the embedding σ : Mn → MI by




σ (c) = c(ı), c .

32

Fix a = σ (a0 ) and b = σ (b0 ), and define B = σ (B0 ). Finally, let h = h0 ◦ σ −1 B .
Construction 3.3.2. [1] Fix m0 an even integer with m0 > |M|. Set k0 = 0 and for each integer
r ≥ 1, set kr = rm0 + 2, so each kr is even. For integers r and i, r ≥ 0 and 0 ≤ i ≤ 1 + kr set
i
qri = 1+k
and define cri ∈ M I as:
r

cr0 = a





a(x), for x ∈ [0, 1 − qri ),




r
ci (x) = b(x), for x ∈ [1 − qr , 1),
i







a(x), for x ≥ 1,

for 1 ≤ i ≤ kr , and

cr1+kr = b.

Note that, for 1 ≤ i ≤ kr the element cri has value a0 (ı) on [0, 1 − qri ) and value b0 (ı) on [1 − qri , 1).
Let Γ be an indexing set for the witnesses of the pp-formula that defines , for γ ∈ Γ let
γ w 0 0 ,γ w 0 0 , and γ w 0 0 be the γ th witnesses for a0  a0 , a0  b0 , and b0  b0 in Mn respectively.
a ,a
a ,b
b ,b

Let γ wa,a = σ (γ wa0 ,a0 ), γ wa,b = σ (γ wa0 ,b0 ), and γ wb,b = σ (γ wb0 ,b0 ). For γ ∈ Γ and 0 ≤ i < j ≤ kr
with j − i even, define γ wri, j ∈ M I by

γ

wri, j (x) =




γ w (x), for x ∈ [0, 1 − qr ),


a,a
j







γ wa,b (x), for x ∈ [1 − qrj , 1 − qri ),


γ w (x), for x ∈ [1 − qr , 1),


b,b

i






γ wa,a (x), for x ≥ 1,

33

and for i odd, with 0 < i < kr , we define γ wi,1+kr ∈ M I by



γ w (x), for x ∈ [0, 1 − qr ),


a,b
i




γ r
wi,1+kr (x) = γ wb,b (x), for x ∈ [1 − qr , 1),
i







γ wa,b (x), for x ≥ 1.
Notice that for 0 ≤ i < j ≤ 1 + kr and j − i even, the element γ wri, j witnesses cri  crj in MI .
For r ≥ 0, let

Cr0 = {cri | 0 ≤ i ≤ 1 + kr },
Wr = {γ wri, j | γ ∈ Γ, 0 ≤ i < j ≤ 1 + kr with j − i even},
Cr = SgMI (Cr0 ∪Wr ),
[

A = SgMI
Cr .
r≥0

In [1] the authors constructed their dense set by taking every homomorphism with domain Cr
for r ≥ 0. Lemma 4.1.2 allows us to build the new dense set using only those homomorphisms with
domain Cr that are irresponsible themselves. Let Pr0 = {g : Cr → M : g(cri ) 6 g(crj ) for some i < j}.
Then, define the set Vh by
Vh = {h} ∪ (

[

Pr0 ).

r≥0

With this construction in hand and Lemmas 3.3.2 and 3.3.3 we have the tools to present our
main theorem: the set Vh is dense-in-FragD(A) .
Theorem 4.2.1. Let M be a finite algebra. Suppose that M has a pp-defined, almost reflexive,
antisymmetric, transitive, binary relation . Suppose further:
34

1. For some B0 ≤ Mn , there exists a homomorphism h0 : B0 → M and there are distinct elements
a0 , b0 ∈ B0 with a0  b0 in Mn , such that h0 (a0 ) 6 h0 (b0 ) and B0 = SgMn ({a0 , b0 }).
2. The associate h : B → M of h0 is a D(A)-fragment, where A, B, and Cr , with r ≥ 0 are all
defined as in Constructions 3.3.1 and 3.3.2.

Then the set Vh is dense in FragD(A) and contains no projections or constant homomorphisms, and
therefore M cannot be strongly dualized.

Proof. Recall that a set of fragments is dense if it is dense at each of its members. We first show
that Vh is dense at h, and then that it is dense for any other fragment, g ∈ Vh .
First, recall h : C0 → M is a D(A)-fragment. Fix r > 0 and recall that C0 ⊆ Cr , assume that
Y ⊆ hom(Cr , M) is a non-empty collection of D(A)-fragments such that h lifts through Y and
|Y | ≤ r. Therefore, there is some homomorphism p : uY (Cr ) → M such that p ◦ (uY )Cr = h.
Assume for a contradiction, that Y ∩ Vh = 0.
/ Then each y ∈ Y is responsible with respect to
cr0  cr1  · · ·  cr1+kr . Therefore, by Lemma 4.1.2, the homomorphism uY partitions {cr1 , . . . , crkr }
into at most
|Y |(|M| − 1) + 1 ≤ r(|M| − 1) + 1
convex sets. Since kr ≥ r(|M| − 1) + 1 we have that

uY (cri ) = uY (crs ) = uY (crj )

for some i < j and all s with i ≤ s ≤ j. Thus, we can choose i0 even and j0 odd with i ≤ i0 ≤ j and
i ≤ j0 ≤ j so that uY (cri0 ) = uY (crj0 ), and cr0  cri0 in Cr and crj0  cr1+kr in Cr . Then, because each

35

y respects cr0  cri0 and crj0  cr1+kr as y 6∈ Vh

uY (cr0 )  uY (cri0 ) = uY (crj0 )  uY (cr1+kr ) in uY (Cr ),

and by applying p we must have

(p ◦ uY )(cr0 )  (p ◦ uY )(cri0 ) = (p ◦ uY )(crj0 )  (p ◦ uY )(cr1+kr ) in M

and by transitivity this gives

h(a) = h(cr0 ) = (p ◦ uY )(cr0 )  (p ◦ uY )(cr1+kr ) = h(cr1+kr ) = h(b)

or more precisely that h(a)  h(b) in M. This is a contradiction, therefore Y ∩ Vh 6= 0/ and Vh is
dense at h.
To prove Vh is dense at each other fragment, pick g ∈ Vh . We know g : Cν → M for some fixed
ν > 0 and that there are i and j with 0 ≤ i < j ≤ 1 + kν such that cνi  cνj but g(cνi ) 6 g(cνj ). Note
that j − i must be odd, otherwise cνi  cνj is witnessed in Cν and so we would have g(cνi )  g(cνj )
witnessed in M.
Again, fix r > 0, and use Lemma 3.3.3 to find integers t and µ such that t is odd, t ≥ kr + 3,
µ

Cν ≤ Cµ and for each 0 ≤ i ≤ 1 + kν , cνi = cti . By the definition of dense at g let Y ⊆ hom(Cµ , M)
be a collection of D(A)-fragments, with |Y | ≤ r and g lifts through Y .
Assume for a contradiction, that Y ∩Vh = 0.
/ Then as before Y is a collection of D(A)-fragments
µ

µ

µ

each of which is responsible with respect to c0  c1  · · ·  c1+kµ . Thus, uY partitions S =
µ

µ

µ

{cti+1 , cti+2 , . . . , ct j−1 } ⊆ Cµ into at most |Y |(|M| − 1) + 1 ≤ r(|M| − 1) + 1 convex sets. So S
has at least (t j − 1) − (ti + 1) = t( j − i) − 2 ≥ t − 2 elements. Now, r(|M| − 1) + 1 < rm0 + 1 <
36

kr < t − 2 so there are i0 , j0 such that uY (ci0 ) = uY (c j0 ) and ti + 1 ≤ i0 < j0 ≤ t j − 1. Therefore,
µ

µ

uY (ci0 ) = uY (cs ) = uY (c j0 ) for all s with i0 ≤ s ≤ j0 . Since i0 < j0 it follows we can choose i∗
µ

µ

µ

and j∗ such that i0 ≤ i∗ ≤ j0 and i0 ≤ j∗ ≤ j0 and both i∗ − ti and t j − j∗ are even and positive. This
µ

µ

µ

µ

µ

µ

µ

µ

gives cti  ci∗ in Cµ and c j∗  ct j in Cµ . Thus, because each y respects cti  ci∗ and c j∗  ct j as
y 6∈ Vh
µ

µ

µ

µ

uY (cti )  uY (ci∗ ) = uY (c j∗ )  uY (ct j ) in uY (Cµ )
If p0 : uY (Cµ ) → M is the map that lifts g through Y , then applying p0 and transitivity to the above
gives:
g(cνi ) = g(cti ) = (p0 ◦ uY )(cti )  (p0 ◦ uY )(ct j ) = g(ct j ) = g(cνj )
µ

µ

µ

µ

This gives g(cνi )  g(cνj ) in M. This is again a contradiction, therefore Y ∩Vh 6= 0/ and Vh is dense
at g.
Since Vh is dense at each of its members, it is dense in FragD(A) . Vh also contains no projections
or constant homomorphisms, therefore, the algebra M is not strongly dualizable by Corollary
3.1.2.

37

Chapter 5

Unary Algebras and Irresponsibility

In the previous chapter we proved a result regarding irresponsible homomorphisms and any algebra, of any type. Since our main focus is with unary algebras, in this chapter we attempt to apply
the previous result to this class of algebras. Throughout this chapter we assume that an algebra M
has a pp-defined binary relation that is almost reflexive, antisymmetric, and transitive; in Section
5.3 we demonstrate that escalator algebras do, in fact, have such a relation.

5.1

Irresponsible Fragments

One useful fact about unary algebras is that the union of subalgebras is again a subalgebra. This
can be seen by looking at the arity of the operations. If A is a unary algebra and B and C are
subalgebras, then A ∪ B is the universe of a different subalgebra. Indeed, since if b ∈ B then for any
operation f in the type of A we have that f (b) ∈ B which implies that f (b) ∈ B ∪C; and similarly,
if c ∈ C then f (c) ∈ C and therefore f (c) ∈ B ∪C.

38

This is important since in Construction 3.3.2 we build an algebra A by taking the union of the
algebras Cr . Since each Cr is a subalgebra of A we can never “pick up” the witness for cri  crj
when j − i is odd in the algebraic closure required to form the algebra A. Therefore, we have the
following.
Corollary 5.1.1. Let M be a finite unary algebra. Suppose that M has a pp-defined binary relation
that is almost reflexive, antisymmetric, and transitive. Suppose further that there exists a subalgebra B of Mn such that there exists a homomorphism h : B → M that is irresponsible with respect
to a  b for some a 6= b, with a, b ∈ B. Then the algebra M cannot be strongly dualized.

This means that finding unary algebras that are not strongly dualizable, provided the algebra
has a pp-defined relation with the required properties, only requires finding irresponsible homomorphisms. Using the main theorem from [1] one still needs to verify that any homomorphism
with domain Cr that happens to be irresponsible is, in fact, irresponsible with respect to cri  crj for
some i < j.

5.2

Algebras with Zero

Suppose that M is a unary algebra with zero and with a pp-defined binary relation, , that is
almost reflexive, antisymmetric, and transitive. Suppose that {(0, 0), (0, c), (c, c)} is contained in
 for some c 6= 0. We claim that the subalgebra generated by the elements (0, 0, c) and (0, c, c) in
M 3 has an irresponsible homomorphism.

39

Let a = (0, 0, c) and b = (0, c, c), let B = SgM3 ({a, b}), and define h : B → M by

h(x) =





π3 (x) x ∈ SgM3 ({a})



π1 (x)

x ∈ SgM3 ({b})

Claim 5.2.1. The map h defined above is an irresponsible homomorphism.

Note that a  b and by definition h(a) = 1 6 0 = h(b) so h is irresponsible. Since both π3 and
π1 are projection maps they are homomorphisms, so we need only show that h is well-defined on
SgM3 ({a}) ∩ SgM3 ({b}).
If x ∈ SgM3 ({a}) ∩ SgM3 ({b}) then x = t(a) and x = s(b) for some terms t and s. Then
(t(0),t(0),t(1)) = (s(0), s(1), s(1)), and so 0 = t(0) = s(1) and we must have that x = (0, 0, 0).
Hence, π3 (x) = π1 (x) = 0 and h is well-defined. Thus, by Corollary 5.1.1 unary algebras with zero
and a pp-defined antisymmetric, transitive, and almost reflexive binary relation where 0  c for
some c 6= 0 are not strongly dualizable.

5.3

Escalator Algebras

For µ ≥ 2 define the algebra Mµ = h{0, 1, . . . , µ}; f , gi, where g(x) = max(0, x − 1) and f (x) =
min(µ, x + 1). Such an algebra is called an escalator algebra and these algebras have been studied
extensively; Figure 5.1 shows the escalator algebra M3 . In [8] it is shown that the three element
escalator algebra is dualizable but not fully dualizable, in [2] it is shown that each escalator algebra
is dualizable and that no escalator algebra is strongly dualizable. In fact, in [1] a new proof using
height is used to prove the same result. We provide this example here because the proof using
Corollary 5.1.1 is shorter than that provided in [1].
40

f
3
g

2

1

0

Figure 5.1: The escalator algebra M3
For µ ≥ 2, define the terms t = gµ−1 f µ−2 and s = gµ−1 f µ−1 on Mµ , where f 0 is the identity.
Then the pp-formula
Φ : ∃ w[t(w) ≈ x & s(w) ≈ y]
defines the almost reflexive, antisymmetric, and transitive relation {(0, 0), (0, 1), (1, 1)}. Then
define a = (0, 0, 1), b = (0, 1, 1), and B = SgM3µ ({a, b}). There is an irresponsible homomorphism
h : B → Mµ determined by h(a) = 1 and h(b) = 0. So we have a  b in M3µ but h(a) 6 h(b) in
Mµ . Therefore, escalator algebras are not strongly dualizable.

5.4

One More Example

Let M5.4 = h{0, 1, 2, 3, 4, 5}, f , gi be the unary algebra defined as in the table below.

41

0
1
2
3
4
5

g
3
4
0
0
1
2

f
3
4
1
0
1
2

Table 5.1: M5.4 = h{0, 1, 2, 3, 4, 5}, f , gi
4
1
5

2
0
3

Figure 5.2: The associated diagram for M5.4
There is an obvious pp-formula,

Φ : ∃ w[x ≈ g(w) & y ≈ f (w)]

which defines the relation = {(0, 0), (0, 1), (1, 1), (2, 2), (3, 3), (4, 4)} that is almost reflexive, antisymmetric, and transitive. In particular we have that 0  1 and the subalgebra B0 = SgM5.4 ({0, 1})
has universe {0, 1, 3, 4}.
Defining h : B0 → M5.4 by h(0) = 1, h(1) = 0, h(3) = 4, h(4) = 3 is indeed a homomorphism.
This claim is easy to verify as f B = gB and h(g(0)) = h(3) = 4 = g(1) = g(h(0)), h(g(1)) =
h(4) = 3 = g(0) = g(h(1)), h(g(3)) = h(0) = 1 = g(4) = g(h(3)), and h(g(4)) = h(1) = 0 = g(3) =
g(h(4)). Further note that this homomorphism is irresponsible as 0  1 in M5.4 but h(0) = 1 6 0 =

42

h(1) in M5.4 . Thus by Corollary 5.1.1 the algebra M5.4 is not strongly dualizable.
There are some properties to note here: First, it has not been investigated in this thesis whether
M5.4 is in fact dualizable and this fact is actually irrelevant in determining that it is not strongly
dualizable, the algebra M5.4 does not have a zero and so is different than those structures studied in
Section 5.2, and the element 2 witnesses 0  1 in M5.4 yet for any term t we have t(0) 6= 2 6= t(1)
an essential part of generating irresponsible homomorphisms.

43

Chapter 6

Conclusion

6.1

Summary

In summary, we have shown that the existence of an irresponsible homomorphism indicates that
an algebra of any type is likely to not be strongly dualizable. In fact, if we can construct, in a
particular way, a subalgebra of a product of the algebra which has a corresponding irresponsible
fragment then the algebra is not strongly dualizable. More specifically, in Chapter 5 we discussed
that the existence of an irresponsible homomorphism directly implies that a unary algebra is not
strongly dualizable. In addition we looked at some cases where irresponsible homomorphisms are
guaranteed to exist. We also provided a new method for constructing dense-in-FragD(A) sets.

44

6.2

Future Work

In this thesis we were able to show that for unary algebras, irresponsibility guarantees that the
algebra is not strongly dualizable. It is possible that future work can refine this process so that
irresponsibility guarantees that an algebra is not strongly dualizable even if that algebra is not
unary. This work would likely be done in steps, perhaps algebras with one binary operation first
and moving on to more complicated structures. Furthermore, within the class of unary algebras
more investigation can be done into what properties are necessary for irresponsibility. Future work
may also investigate if irresponsibility with respect to other relations also indicates that an algebra
is not strongly dualizable.
One problem that occurred in research was with meet homomorphisms. A meet homomorphism provides a natural way to define an ordering on the universe of the algebra which may or
may not agree with the pp-defined order . In the case where the two orders agree (where they
are defined) it seems likely that an irresponsible homomorphism must exist, however, we were not
able to prove that either

a) The order defined by the meet homomorphism must agree with the pp-defined order , or
b) If the orders disagree whether irresponsibility is guaranteed or not.

The algebra in Section 5.4 does not have a meet homomorphism, so this provides a perfect
example that meet homomorphisms are not the only place to look despite being interesting. This
example suggests that we should look to find easily identifiable structure that guarantees that subalgebras do not pick up undesired witnesses for the pp-defined relation.

45

List of Symbols
B≤A
ISP(M)
ISc P+ (M)
SP(M)
BA
FragD(A)
SgA (B)

B is a subalgebra of A
The class of all isomorphic copies of subalgebras of products of M
The class of all isomorphic copies of closed substructures of positive products of M
The class of all subalgebras of products of M
B is a finite subalgebra of A
The set of all D(A)-fragments
The subalgebra of A generated by the set B
Table 1: A list of important symbols used in this thesis

46

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