Universal algebra

233 Pages·2008·1.533 MB·English

by  Jaroslav Jezek

UNIVERSAL ALGEBRA
Jaroslav Jeˇzek
First edition, April 2008

Contents
PREFACE 1
Chapter 1. SET THEORY 3
1. Formulas of set theory 3
2. Theory of classes 5
3. Set theory 8
4. Relations and functions 10
5. Ordinal numbers 12
6. Cardinal numbers 18
Comments 21
Chapter 2. CATEGORIES 23
1. Basic definitions 23
2. Limits and colimits 24
3. Complete and cocomplete categories 26
4. Reflections 28
Chapter 3. STRUCTURES AND ALGEBRAS 29
1. Languages, structures, algebras, examples 29
2. Homomorphisms 33
3. Substructures 34
4. Congruences 36
5. Direct and subdirect products 37
6. ISP-closed classes 39
7. Free partial structures 40
8. The category of all partial structures of a given language 41
9. ISP-closed classes as categories 42
10. Terms 44
11. Absolutely free algebras 45
12. Representation of lattices by subuniverses and congruences 46
Chapter 4. LATTICES AND BOOLEAN ALGEBRAS 53
1. Modular and distributive lattices 53
2. Boolean algebras 55
3. Boolean rings 57
4. Boolean spaces 57
5. Boolean products 60
iii
iv CONTENTS
Chapter 5. MODEL THEORY 63
1. Formulas 63
2. Theories 65
3. Ultraproducts 65
4. Elementary substructures and diagrams 66
5. Elementary equivalence 68
6. Compactness theorem and its consequences 69
7. Syntactic approach 70
8. Complete theories 73
9. Axiomatizable classes 74
10. Universal classes 75
11. Quasivarieties 76
Chapter 6. VARIETIES 79
1. Terms: Syntactic notions 79
2. The Galois correspondence 80
3. Derivations, consequences and bases 82
4. Term operations and polynomials 82
5. Locally finite and finitely generated varieties 84
6. Subdirectly irreducible algebras in varieties 85
7. Minimal varieties 85
8. Regular equations 87
9. Poor signatures 91
10. Equivalent varieties 91
11. Independent varieties 93
12. The existence of covers 94
Chapter 7. MAL’CEV TYPE THEOREMS 97
1. Permutable congruences 97
2. Distributive congruences 100
3. Modular congruences 101
4. Chinese remainder theorem 105
5. Arithmetical varieties 107
6. Congruence regular varieties 108
7. Congruence distributive varieties 109
8. Congruence meet-semidistributive varieties 110
Chapter 8. PROPERTIES OF VARIETIES 113
1. Amalgamation properties 113
2. Discriminator varieties and primal algebras 117
3. Dual discriminator varieties 121
4. Bounded varieties 124
Chapter 9. COMMUTATOR THEORY AND ABELIAN ALGEBRAS 129
1. Commutator in general algebras 129
2. Commutator theory in congruence modular varieties 131
CONTENTS v
3. Abelian and Hamiltonian varieties 133
Chapter 10. FINITELY BASED VARIETIES 137
1. A sufficient condition for a finite base 137
2. Definable principal congruences 137
3. J´onsson’s finite basis theorem 139
4. Meet-semidistributive varieties 140
Comments 144
Chapter 11. NONFINITELY BASED VARIETIES 145
1. Inherently nonfinitely based varieties 145
2. The shift-automorphism method 146
3. Applications 149
4. The syntactic method 151
Comments 152
Chapter 12. ALGORITHMS IN UNIVERSAL ALGEBRA 153
1. Turing machines 153
2. Word problems 155
3. The finite embedding property 157
4. Unsolvability of the word problem for semigroups 159
5. An undecidable equational theory 161
Comments 161
Chapter 13. TERM REWRITE SYSTEMS 163
1. Unification 163
2. Convergent graphs 165
3. Term rewrite systems 166
4. Well quasiorders 168
5. Well quasiorders on the set of terms 170
6. The Knuth-Bendix algorithm 171
7. The Knuth-Bendix quasiorder 172
8. Perfect bases 174
Chapter 14. MINIMAL SETS 183
1. Operations depending on a variable 183
2. Minimal algebras 184
3. Minimal subsets 187
Chapter 15. THE LATTICE OF EQUATIONAL THEORIES 197
1. Intervals in the lattice 197
2. Zipper theorem 199
Chapter 16. MISCELLANEOUS 201
1. Clones: The Galois correspondence 201
2. Categorical embeddings 209
OPEN PROBLEMS 219
vi CONTENTS
References 221
Index 225
PREFACE
This is a short text on universal algebra. It is a composition of my various
notes that were collected with long breaks for many years, even decades; re-
cently I put it all together to make it more coherent. Some parts were written
and offered to my students during the past years.
The aim was to explain basics of universal algebra that can be useful for a
starting advanced student, intending possibly to work in this area and having
somebackgroundinmathematicsandinalgebrainparticular. Iwillbeconcise.
Many proofscould beconsidered as justhints for proving theresults. Thetext
could be easily doubled in size. Almost no mention of the history of universal
algebra willbegiven; sufficeitto say that foundationswere laid by G. Birkhoff
in the 1930’s and 1940’s. There will be not many remarks about motivation or
connections to other topics. We start with two chapters collecting some useful
knowledge of two different subjects – set theory and the theory of categories,
just that knowledge that is useful for universal algebra. Also, the chapter on
model theory is intended only as a server for our purposes.
The bibliography at the end of the book is not very extensive. I included
only what I considered to be necessary and closely related to the material
selected for exposition. Many results will be included without paying credit to
their authors.
Selectionofthetopicswasnotgivenonlybymyesteemoftheirimportance.
The selection reflects also availability provided by my previous notes, and
personal interest. Some most important modern topics will not be included,
or will be just touched. This text contains no original results.
Idonotkeeptotheusualconvention ofdenotingalgebrasbyboldfacechar-
acters and then their underlying sets by the corresponding non-bold variants.
Instead, I use boldface characters for constants (of any kind) and italics, as
well as greek characters (both upper- and lower-case), for variables (running
over objects of any kind). I do not reserve groups of characters for sorts of
variables.
Therarecases whenitis really necessarytodistinguishbetween analgebra
and its underlying set, can be treated by adding a few more words.
Other texts can be also recommended for alternative or further reading:
Burris and Sankappanavar [81]; McKenzie, McNulty and Taylor [87]; Hobby
and Mckenzie [88]; Freese and McKenzie [87]; Gr¨atzer [79]; Garc´ıa and Taylor
[84]; Gorbunov [99]. Some material in the present book has been also drawn
from the first four of these books.
1
2 PREFACE
Iwouldbegratefulforcommentsofanykind,andinparticularforpointing
outerrors or inconsistencies. I could usethem for improvements that would be
included in a possible second edition which may also contain some extensions.
Please contact me at [email protected]ff.cuni.cz.
My thanks are to Ralph McKenzie, Miklos Mar´oti and Petar Markovi´c for
many friendly discussions that were also of great value when I was working on
this text.
CHAPTER 1
SET THEORY
The whole of mathematics is based on set theory. Because intuitive set
theory can easily lead to the well-known paradoxes (the set A of all the sets
that are not their own elements can satisfy neither A ∈ A nor A ∈/ A), it is
reasonableto work inatheory with acarefully selected system of axioms. Two
such axiom systems, essentially equivalent, are the most common: the G¨odel-
Bernays and the Zermelo-Fraenkel systems. For universal algebra the first is
the more convenient. The purpose of this chapter is to present foundations of
settheorybasedontheG¨odel-Bernayssystemofaxioms. ThebooksG¨odel[40],
Cohen [66] and Vopˇenka, H´ajek [72] can be recommended for further reading.
1. Formulas of set theory
Certain stringsofsymbolswillbecalled formulas. Symbolsthatmay occur
in the strings are the following:
(1) Variables: both lower- and upper-case italic letters or letters of the
Greek alphabet, possibly indexed by numerals (there should be no
restriction on the number of variables)
(2) One unary connective: ¬
(3) Four binary connectives: & ∨ → ↔
(3) Two quantifiers: ∀ ∃
(4) Parentheses: ( )
(5) Equality symbol: =
(6) Membership symbol: ∈
A formula is a string that can be obtained by several applications of the
following rules. For every formula we also specify which variables are called
free and which variables are called bound in it.
(1) For any two (not necessarily distinct) variables x and y, the strings
x=y and x∈y are formulas; both x and y are free, no other variable is
free, and no variable is bound in these formulas.
(2) If f is a formula then ¬(f) is a formula; a variable is free (or bound)
in ¬(f) if and only if it is free (or bound, respectively) in f.
(3) If f and g are two formulas and if no variable is either simultaneously
free in f and bound in g or simultaneously bound in f and free in g,
then the four strings (f)&(g) and (f)∨(g) and (f)→(g) and (f)↔(g)
are formulas; a variable is free (or bound) in the resulting formula if
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4 1. SET THEORY
and only if it is free (or bound, respectively) in at least one of the the
formulas f and g.
(4) If f is a formula and if x is a variable that is not bound in f, then
both (∀x)(f) and (∃x)(f) are formulas; the variable x is bound in the
resulting formula; a variable other than x is free (or bound) in the
resulting formula if and only if it is free (or bound, respectively) in f.
Observethatnovariable isbothfreeand boundin any formula. Avariable
occurs in a formula if and only if it is either free or bound in it. By a sentence
we mean a formula without free variables.
Certain formulas are called logical axioms. If f, g and h are three formulas,
then the following are logical axioms provided that they are formulas (some
parentheses are omitted):
(1) f→(g→f)
(2) (f→(g→h))→((f→g)→(f→h))
(3) ((¬f)→(¬g))→(g→f)
(4) (f↔g)→(f→g)
(5) (f↔g)→(g→f)
(6) (f→g)→((g→f)→(f↔g))
(7) (f∨g)↔((¬f)→g)
(8) (f&g)↔¬((¬f)∨(¬g))
(9) ((∀x)f)→g where x and y are two variables not bound in f and g is
obtained from f by replacing all the occurrences of x with y
(10) ((∀x)(f→g))→(f→(∀x)g) where x is a variable not occurring in f
(11) ((∃x)f)↔¬((∀x)¬f) where x is a variable not bound in f
(12) x=x where x is a variable
(13) x=y→y=x where x and y are two variables
(14) (x=y&y=z)→x=z where x, y and z are three variables
(15) (x=y&z=u)→(x∈z↔y∈u) where x, y, z and u are four variables
By a theory we mean a (finite) collection of formulas of the language; these
formulas are called axioms of that theory.
By a proof in a given theory T we mean a finite sequence of formulas such
that each member of the sequence is either a logical axiom or an axiom of T
or can be obtained from one or two earlier members of the sequence by one of
the following two rules:
(1) obtain g from f and f→g;
(2) obtain f from (∀x)f.
By a proof of a given formula in a given theory T we mean a proof in
T which has the given formula as its last member. A formula is said to be
provable in T if there exists a proof of the formula in T.
A theory S is said to be an extension of a theory T if every axiom of T
is an axiom of S. A theory S is said to be stronger than a theory T if every
axiom of T is provable in S. As it is easy to see, it follows that each formula
provable in T is also provable in S. Two theories are said to be equivalent if