Subdirectly irreducible algebras in varieties of universal algebras

Transcription

1 Subdirectly irreducible algebras in varieties of universal algebras Dorien Zwaneveld August 5, 204 Bachelorthesis Supervisors: dr. N. Bezhanishvili, J. Ilin Msc Korteweg-de Vries Institute for Mathematics Institute for Logic, Language and Computation Faculty of Science University of Amsterdam

2 Abstract In this thesis we give a characterization of subdirectly irreducible algebras within the varieties of Boolean algebras and of Heyting algebras. In order to do this we will start with an introduction into universal algebra and in particular in lattice theory. We will define congruences and we will define varieties via the operations of homomorphic images, subalgebras and products. Then we will prove Birkhoff s Theorem which states that a class of algebras is an equational class if and only if it is a variety. In the last chapter we will define subdirectly irreducible algebras. We will give characterizations of subdirectly irreducible universal algebras via congruences. For subdirectly irreducible Boolean and Heyting algebras we will work out a more convenient characterization via filters. As a result we obtain a direct characterization of subdirectly irreducible Boolean and Heyting algebras. Title: Subdirectly irreducible algebras in varieties of universal algebras Author: Dorien Zwaneveld, tcam.zwaneveld@gmail.com, 6457 Supervisors: dr. N. Bezhanishvili, J. Ilin Msc Second Grader: Prof. dr. Y. Venema Deadline: August 5, 204 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park 904, 098 XH Amsterdam Institute for Logic, Language and Computation University of Amsterdam Science Park 07, 098 XG Amsterdam 2

3 Acknowledgements During my research for this thesis I was an intern at a high school, on my way to becoming a teacher, like most of the people who contributed to the theory discussed in this thesis. This gave me a lot of stress and therefore I would like to thank my boyfriend, Jelle Spitz, for supporting me, loving me and keeping faith even when I went crazy and did not have much faith myself. I would like to thank my teachers and friends at the ILO for supporting me and Chris Zaal for cutting me some slack during these stressful times. I would also like to thank my friends, Hanneke van der Beek and Timo de Vries for getting up early so many times during the summer and supporting me in writing this thesis. Furthermore I would like to thank my parents and sisters for their love and support. Most of all I give many thanks to my supervisors Nick Bezhanishvili and Julia Ilin, whose doors were always open and who pointed me in the right direction when I got stuck. Without these people I would have never made it this far. 3

4 Contents Introduction 5 2 An introduction to universal algebra 6 2. Universal algebras Lattice theory Boolean and Heyting algebras Varieties of universal algebras 7 3. Congruences Varieties Homomorphisms Subalgebras Products Tarski s Theorem Term algebras Identities The K-free algebra Subdirectly irreducible algebras Building blocks of varieties Determining subdirectly irreducible algebras Conclusions 53 6 Popular summary 54 Bibliography 56 4

5 Introduction During a typical bachelors in mathematics one learns about groups and rings and it gets called algebra. But what is an algebra? A question raised by many mathematicians during the 9th century, but it took until 933 that the first definition of a universal algebra was given. During the studies of algebraic structures (or algebras) one finds that certain theorems do not only hold for groups, but they also hold for rings and for other structures as well. (Think of the homomorphism theorem for example.) In universal algebra one does not only look at groups, or rings, or..., one looks at the whole picture. In this approach one is taking a step back and is looking at things from a distance. It underlines general patterns that various algebraic structures have in common. This is something logicians do more often than mathematicians. Since this thesis is written at the ILLC, taking a step back and looking at the whole picture is what I will be doing in the first part of my thesis. I will be looking at the concept of universal algebra and in particular lattice theory. Universal algebra is a relatively new part of mathematics founded by Garrett Birkhoff (9-996) during the 930s when he was teaching at the Harvard University. Garrett Birkhoff did not have a doctoral degree or even a masters degree, but still he is one of the main influences in the mathematical branch called universal algebra. He was the one who gave the definition of a universal algebra and he also contributed greatly to lattice theory which is why we will come by his name fairly often in this thesis. We will also take a look at some algebraic structures that are less familiar to mathematicians, but more familiar to logicians: Boolean algebras and Heyting algebras. Boolean algebras are named after George Boole (85-864), who was an English mathematician as well as logician and philosopher. Boolean algebras are models for classical propositional logic. Because of this, Boolean algebras have applications in computer science that turn out to be very useful. At the end of the thesis we briefly discuss the connection between Boolean algebras and classical propositional logic and we will see that the 2-element Boolean algebra plays a special role for classical logic. We will also discuss Heyting algebras and show that every Boolean algebra is a Heyting algebra. These algebras were named after a Dutch mathematician and logician: Arend Heyting ( ). Heyting s calculus formalizes the ideas of the Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer (88-966) on constructive mathematics. They both studied and taught mathematics at the University of Amsterdam. In short, Heyting algebras are to intuitionistic propositional logic what Boolean algebras are to classical propositional logic. Boolean and Heyting algebras form varieties (as do groups and rings). In this thesis we will take a closer look at these algebraic structures with the main goal as to give a characterization of the building blocks for the varieties of Heyting and Boolean algebras. 5

6 2 An introduction to universal algebra The goal of this first chapter is to define and develop an understanding of varieties. To do this, we will start with a universal algebra. What is a universal algebra? Do we already know some examples? We will answer these question and then we will move on to lattice theory. This is an area of logic and universal algebra one is unlikely to encounter in a bachelors in mathematics. It is also very important for the contents of this thesis, because this is one of the major building blocks for the rest of the theory. After discussing the basics of lattice theory, we will turn to some examples of algebras arising from logic: Boolean algebras and Heyting algebras. 2. Universal algebras The reader may have encountered examples of algebras in various contexts. But in most of these contexts we often do not know what an algebra actually is. So let us start with giving a definition of an algebra. But first let us define the type of an algebra. Definition 2.. A type of algebras is a set F of function symbols. Each of these function symbols has a nonnegative integer n assigned to it. This integer is called the arity of the symbol and we say that the function symbol is n-ary. Definition 2.2. A (universal) algebra A of type F is an ordered pair A, F where A is a nonempty set and F is a family of finitary operations on A indexed by the type F such that every n-ary function symbol corresponds to an n-ary operation on A. We call the set A the underlying set of A. Note that the set A is closed under the finitary operations in F. So the operations in an algebra are the interpretation of the function symbols of the type. Now let us look at some examples of universal algebras. The reader may have seen some of these before. Example 2.3. Groups A group is a set G together with the binary operation, the unary operation and the nullary operation. Notation: G,,,. Rings A ring R, +,,, 0 is a set R with two binary operations + and, one unary operation and one nullary operation 0. 6

7 Sigma-algebras Also studying the mathematical field of measure theory one encounters algebras. Given a set Σ of elements one can define different sigma-algebras. For let S be the some collection of subsets of Σ, then S,,, is one of the sigma-algebras on Σ. Lattices This is an algebra we have not yet seen before. A lattice has two binary operations L,,. Lattices are important for the contents of this thesis and therefore we will look at them more closely. Boolean algebras A Boolean algebra is a lattice together with some extra properties and operations B,,,, 0,. We will find out more about these algebras further in this thesis. Heyting algebras We will also discuss some properties and examples of Heyting algebras. A Heyting algebra is also a lattice with the same type as Boolean algebras, but they satisfy different axioms H,,,, 0,. In these algebras there are usually finitely many operations, we put the operation with the biggest arity first and the operation with the smallest arity last. Definition 2.4. Let G and G be two groups. A map f : G G is called a homomorphism if for all x, y G (i) f(x y) = f(x) f(y); (ii) f(x ) = f(x) ; (iii) f() =. Here is the nullary operation in G. For two rings R and R, a mapping f : R R is called a homomorphism if for all x, y R we have (i) f(x + y) = f(x) + f(y); (ii) f(x y) = f(x) f(y); (iii) f( x) = f(x); (iv) f(0) = 0. As with the groups, 0 is the nullary operation in R. Now we will generalize these to any universal algebra. 7

8 Definition 2.5. When we have two algebras A and B of the same type, a mapping α : A B is called a homomorphism if α(f A (a,..., a n )) = f B (α(a ),..., α(a n )). Here a,..., a n are elements in A and f is an n-ary operation. f A is the interpretation of f on A. Now that we have the concept of a homomorphism it is convenient to extend this idea and specify a couple of different kinds of homomorphisms. Definition 2.6. An injective (one-to-one) homomorphism is called an embedding or a monomorphism. When a homomorphism is surjective (onto) it is an epimorphism. A homomorphism that is both injective and surjective is called an isomorphism. With these homomorphisms we prove some results later on. 2.2 Lattice theory In the previous section we have seen that a lattice is an algebra. In this section we will see some different definitions of a lattice and we will prove that these definitions are indeed equivalent. We will also look at some special lattices and some of their properties. We will start by giving a definition of a lattice as given in the first paragraph. Definition 2.7. A lattice is an algebra L,, with two binary operations which satisfy the identities: (commutative laws) (associative laws) (idempotent laws) x y y x x y y x x (y z) (x y) z x (y z) (x y) z x x x x x x (absorption laws) x x (x y) x x (x y) The binary operation is also called the meet -operation and is called the join. Since a lattice is also an algebra it is implicit that our set L is non-empty. The reader may have seen these operation symbols before, namely in the propositional logic. There is called and or conjunction and is called or or disjunction. When we look at it this way, the properties from propositional logic form the laws. We model this by taking the propositions as elements of a set and note that this actually forms a lattice. Recall that a partially ordered set is a set with a partial order on it. A partial order is an ordering which is reflexive, antisymmetric and transitive. We can use partials orders to get another definition of a lattice. 8

9 Definition 2.8. A lattice is a partially ordered set L for which the following holds: a, b L : sup{a, b}, inf{a, b} L. Here sup{a, b} is the least upper bound of a and b. Thus sup{a, b} = p a p, b p and for all c such that a c and b c we have p c. Here is the partial order on the set. The infimum is the greatest lower bound i.e. inf{a, b} = p a p, b p and for all c such that a c and b c we have p c. So a partially ordered set L is a lattice if the supremum and infimum of every 2-element set exist in L. We have now seen two different definitions of a lattice. But are they in fact equivalent? We will show this in the following theorem. Theorem 2.9. Definition 2.7 and Definition 2.8 are equivalent. Proof. Suppose that L is a lattice by Definition 2.7. We have to create a partial order on L. Define: a b a = a b. Now all we have to do is show that the -relation defined above is a partial order and that by this definition the supremum and infimum of every 2-element set exist. By the idempotent law we find that a = a a and therefore we have established reflexivity. Now suppose we have a b and b a. We find a = a b = b a = b. Thus by the commutative law and the above we also have antisymmetry. For the transitivity, suppose a b and b c. The above tells us that a = a b and b = b c thus a = a (b c). Now we can use the associative law to find that a = (a b) c. But a = a b so this tells us that a = a c. Therefore we have a c and we have proved the transitivity. Now suppose that a and b are two elements from L. We want to find an element p such that p = sup{a, b}. Since by absorption a (a b) = a and b (a b) = b holds, we find a a b and b a b. Thus a b is an upper bound for both a and b. Now suppose there exists some element c such that a c and b c. Then a c = a and b c = b so a c = (a c) c = c and b c = (b c) c = c by absorption again. Now (a b) c = (a b) (c c) (idempotent) = (a c) (b c) (associative) = c c = c Thus we find (a b) c = (a b) ((a b) c) = a b by using absorption. Therefore a b c and sup{a, b} = a b. Now we use the fact that a lattice is an algebra and therefore L is closed under thus sup{a, b} exists in L. We also want to find an element q such that q = inf{a, b}. Since sup{a, b} = a b it seems to be a rather safe assumption that a b = inf{a, b}. So let us see why this is true. (a b) a = a b by associativity. In the same way we have (a b) b = a b. 9

10 So we find a b a and a b b. Now assume there exists some element d such that d a and d b. Then a d = d = b d. Now (a b) d = (a b) (d d) (idempotent) = (a d) (b d) (associative) = d d = d Thus we find d a b and therefore inf{a, b} = a b as we claimed. So our first definition (Definition 2.7) implies our second definition (Definition 2.8). Now let us assume that we have a partially ordered set L such that for every 2-element set {a, b} L their infimum and supremum exist within L. For a, b L define sup{a, b} = a b and inf{a, b} = a b. a b = inf{a, b} = inf{b, a} = b a a b = sup{a, b} = sup{b, a} = b a a (b c) = inf{a, inf{b, c}} = inf{a, b, c} = inf{inf{a, b}, c} = (a b) c a (b c) = sup{a, sup{b, c}} = sup{a, b, c} = sup{sup{a, b}, c} = (a b) c a a = inf{a, a} = a a a = sup{a, a} = a a (a b) = inf{a, sup{a, b}} = a a (a b) = sup{a, inf{a, b}} = a (commutative laws) (associative laws) (idempotent laws) (absorption laws) So the partially ordered set satisfies the identities from Definition 2.7. It becomes an algebra by noticing that L is also closed under and since the supremum and infimum are in L for every 2-element set. Thus the definitions are indeed equivalent. In Figure 2. one will find some examples of lattices while in Figure 2.2 one will find a picture of a Hasse diagram that is not a lattice. This is because sup{c, d} and inf{a, b} do not exist. By Definition 2.8 we deduce that Figure 2.2 is not a lattice. 0

11 a = a b a = a c = a c a b a b b = a b = b c a b a b c b = a b c = a c = b c a b a c a b a b a b b c a b a b c d a b a c a b c d a b a b c b (a b) c c d d a b Figure 2.: some lattices a b c d Figure 2.2: not a lattice Definition 2.0. A lattice is called distributive if it satisfies one of the distributive laws: (distributive laws) x (y z) (x y) (x z) x (y z) (x y) (x z) Definition 2.. A lattice is called modular if it satisfies the modular law: (modular law) x y x (y z) y (x z) In Figure 2.3 one of the lattices is modular but not distributive and the other lattice is neither modular nor distributive. These two lattices are called M 5 and N 5, respectively. We will use these to characterize non-modular and non-distributive lattices.

12 0 y x y z x 0 (a) M 5 (b) N 5 z and Figure 2.3 To see that M 5 is not distributive let us take x, y and z as in Figure 2.3a. Now x (y z) = x = x (x y) (x z) = 0 0 = 0 therefore x (y z) (x y) (x z) and we deduce that M 5 is not distributive. But M 5 is modular, there are ten cases to distinguish, and checking them takes a little time but is routine. As stated before, N 5 is not distributive, and not modular. To see this, let us take x, y and z as in Figure 2.3b. We find x (y z) = x 0 = x y = y = (x y) (x z) therefore N 5 cannot be distributive. Also, x y so for N 5 to be modular we should have x (y z) = y (x z). But we have x (y z) = x 0 = x y = y = y (x z). Thus N 5 is neither a modular nor distributive lattice. This gives rise to a question: does the implication distributive modular hold for every lattice? Theorem 2.2. Any distributive lattice is modular. Proof. Suppose L is a distributive lattice and x, y, z L. Then x y means x y = x but we have also seen that it means x y = y. Now the distributive law states x (y z) = (x y) (x z) = y (x z). And thus we have satisfied the modular law. The next theorem proved by R. Dedekind gives an easy way to identify modular lattices using N 5 as in Figure 2.3. He also uses the fact that an embedding of a lattice L into another lattice L can be seen as if L contains a copy of L where the lattice operations of L hold. Theorem 2.3 (Dedekind). L is a nonmodular lattice if and only if N 5 can be embedded into L. 2

13 Proof. [, Theorem 3.5] G. Birkhoff used this theorem to prove another theorem which gives a characterization of distributive lattices. Since we will be looking at these lattices, we will state the theorem here. Theorem 2.4 (Birkhoff). L is a nondistributive lattice if and only if M 5 or N 5 can be embedded into L. Proof. [, Theorem 3.6] 2.3 Boolean and Heyting algebras Now that we introduced distributive lattices, let us look at some special distributive lattices: Boolean and Heyting algebras. We will start this section with the definitions and then we will discuss some properties of these algebras. At the end of the thesis we will look at classes of Boolean and Heyting algebras. Definition 2.5. A Boolean algebra is an algebra B,,,, 0, with two binary, one unary, and two nullary operations which satisfy: B,, is a distributive lattice x 0 0; x x x 0; x x. In short, a Boolean algebra is a distributive lattice with a top element and a bottom element 0 and where every element has a unique complement. In Figure 2.4 some examples are shown. 0 a a 0 Figure 2.4: Some Boolean algebras a b c c b a 0 Now that we have seen some Boolean algebras, let us define Heyting algebras. Definition 2.6. An algebra H,,,, 0, with three binary and two nullary operations is a Heyting algebra if it satisfies: (i) H,, is a distributive lattice 3

14 (ii) x 0 0; x (iii) x x (iv) (x y) y y; x (x y) x y (v) x (y z) (x y) (x z); (x y) z (x z) (y z). The binary operation sends two elements x and y to the element x y. In Figure 2.5 we determine this element for some cases. 0 = 0 x = x 0 x = y x y = x 0 0 w x = y z y = x 0 z 0 Figure 2.5: Some Heyting algebras 0 0 When determining a b when given two elements a and b the following theorem can be very useful. To prove it we first need some auxiliary lemmas. Lemma 2.7. In any Heyting algebra the following holds: (i) x (y x) x y (ii) a b implies x a x b. Proof. (i) We have x (y x) (x y) (x x) (by (v) of 2.6) (x y) (by (iii) of 2.6) x y (Since for all a we have a thus a = a) (ii) Suppose a b then we have that x a x (a b) since a a b follows from the fact that a b. Now by (v) of Definition 2.6 it follows that a b implies (x a) (x b). 4

15 With these properties we can prove the following theorem. Theorem 2.8. If H,,, 0, is a Heyting algebra and a, b H then a b is the largest element c of H such that a c b. Proof. Let us start with the easy part. One of the properties of a Heyting algebra is x (x y) x y. So the first part follows directly from the properties of the Heyting algebra. Now we use the previous lemma to prove the other direction. So let us assume that (a c) b. Then by Lemma 2.7 (ii) a (a c) (a b). But by Lemma 2.7 (i) this means that (a c) (a b). Now, by the same property of the Heyting algebra we used before, we find c (a b). Thus a b is indeed the largest element c in H such that a c b. Lemma 2.9. If H,,, 0, is a Heyting algebra and a, b H then a b if and only if a b =. Proof. If a b then a = a b. Since (a b) = a b we have that (a b) b. Since is the biggest element in H we find by Theorem 2.8 that a b =. Another way to prove this is the following: a b = (a b) = (a b) (a a) = a (b a) = a a = Now suppose that a b = then again by Theorem 2.8 we find that a = a b. In a Heyting algebra the infinite distributive law also holds as we see in the following lemma. We will need this result later in the thesis when we determine subdirectly irreducible Heyting algebras. Lemma For any Heyting algebra A, a set S A and x A, the following holds: x S = {x s s S}. Proof. The proof of the lemma can be found in [2, Proposition 2.2.7]. As the reader may have noticed, examples of Boolean algebras are also examples of Heyting algebras. This is a fact that holds for every Boolean algebra. Lemma 2.2. Let B,,,, 0, be a Boolean algebra. Heyting algebra with a b = a b for a, b B. Then B,,,, 0, is a 5

16 Proof. We have to check the five axioms (or identities as we will call them later) that hold for any Heyting algebra. Suppose x, y, z B then we can say the following: (i) It follows from the definition of a Boolean algebra that B,, is a distributive lattice. (ii) x 0 0 and x are also axioms of Boolean algebras, so they hold. (iii) x x = x x =. This follows from the definition of in a Boolean algebra. (iv) Since B,, is a distributive lattice, we can use the absorption law to find (x y) y = (x y) y = y. Now for the other part, we use the distributivity of the lattice and the definition of the complement and the nullary operation 0 to see that x (x y) = x (x y) = (x x ) (x y) = 0 (x y) = x y. (v) Again from the distributivity we have x (y z) = x (y z) = (x y) (x z) = (x y) (x z). The last part of the last property uses the next claim. Claim: In a Boolean algebra B : (x y) x y. To see this, just checking the properties of the complement in a Boolean algebra is enough. So suppose x, y B then we have (x y) (x y ) = ((x y) x ) ((x y) y ) and we obtain = ((x x ) y) (x (y y )) =. (x y) (x y ) = (x (x y )) (y (x y )) = ((x x ) y ) (x (y y )) = 0. So now we can prove the last property of the Heyting algebras. This prove uses the claim as stated and it uses the distributivity of the lattice again: (x y) z = (x y) z = (x y ) z = (x z) (y z) = (x z) (y z). 6

17 3 Varieties of universal algebras We have introduced Boolean and Heyting algebras and we have discussed some of their properties. So now let us take a step back again and go back to universal algebra. In this chapter we will define the notion of varieties. The goal of this chapter is to obtain two characterizations of varieties, proved by Alfred Tarski and by Garrett Birkhoff, respectively In order to do this we will define congruences in the first section and we will discuss the K-free algebra with some of its properties in the last. 3. Congruences In this section we will discuss the notion of congruences. We will define them and we will discuss some of their properties. We will see that all congruences on a universal algebra form a lattice where is the order. We will also give some examples of congruence lattices on Heyting algebras. We will end this section by defining the quotient algebra. Definition 3.. An equivalence relation on a set V is a subset R of V V such that for all x, y, z V : (reflexivity) (x, x) R; (symmetry) If (x, y) R then (y, x) R; (transitivity) If (x, y) R and (y, z) R then (x, z) R. Definition 3.2. Let A be an algebra. An equivalence relation θ on A is a congruence if for every n-ary fundamental operation of A and a,..., a n, b,..., b n A the following holds: a i, b i θ = f A (a,..., a n ), f A (b,..., b n ) θ. Let us denote the set of all congruences on A by Con(A). So a congruence on an algebra A is an equivalence relation on A with the extra property that a congruence is compatible with the operations of A. Now let us define the meet and join of two congruences in order to deduce that the set of all congruences Con(A) forms a lattice. Definition 3.3. For two congruences θ and φ on an algebra A let us define θ φ := θ φ θ φ := θ (θ φ) (θ φ θ) (θ φ θ φ).... 7

18 Here is the relational product defined by a, b θ φ c A : a, c θ, c, b φ. Then Con(A) = Con(A),, is the congruence lattice of A. Every congruence lattice has a largest and smallest element. These are the trivial congruences on the algebra. On an algebra A, the smallest congruence is = { a, a A 2 } and the largest one is = A A. In Figure 3. we show some of the Heyting algebras we have seen in the previous section together with their congruence lattices. 0 x 0 θ x y θ 2 θ 3 0 x y θ 4 z θ 5 θ 6 0 8

19 x θ 7 θ 8 z y θ 9 0 x w θ 0 θ z y θ 2 θ 3 0 Figure 3.: Some Heyting algebras on the left with their congruences lattices on the right. In Figure 3. θ to θ 3 are defined as in Table 3.. θ {, x, x, } θ 2 θ 3 {, x, x,, y, 0, 0, y } {, y, y,, x, 0, 0, x } θ 4 { a, b a, b z} θ 5 {, x, x,, y, z, z, y } {, y, y,, x, z, z, x } θ 6 θ 7 { a, b a, b y} { z, 0, 0, z } θ 8 { a, b a, b z} { y, 0, 0, y } θ 9 {, x, x, } θ 0 { a, b a, b y} { c, d c, d x} θ { a, b a, b z} { y, 0, 0, y } θ 2 {, w, w,, x, z, z, x } {, x, x,, w, z, z, w } θ 3 Table 3. We know that an equivalence relation on a group or ring gives rise to a quotient. Since a congruence is an equivalence relation this also gives rise to a quotient. We will generalize this concept and define the quotient algebra via a congruence: 9

20 Definition 3.4. The quotient algebra of A by θ (notation: A/θ) is the algebra with A/θ, the set of all congruence classes of A by θ, as its underlying set. For any element a A, we denote the congruence class of a via θ by a/θ. The operations of the quotient algebra satisfy: f A/θ (a /θ,..., a n /θ) = f A (a,..., a n ). Here a,..., a n A and f is an n-ary operation of A. Let us end this section with a useful property of the congruence, the proof of this lemma is straightforward see, e.g. the book A Course in Universal Algebra by Stanley Burris and H.P. Sankappanavar [, Theorem 5.9]. Lemma 3.5. Let A be an algebra and suppose θ, φ Con(A). Then the following are equivalent: (i) θ φ = φ θ (ii) θ φ = φ θ (iii) θ φ φ θ. 3.2 Varieties In this section we will take another look at homomorphisms, discuss some of their properties and define homomorphic images. Then we will define the notion of a subalgebra and after that we will define products of algebras. We will need these three notions in order to give a definition of a variety. We will end this section with Tarski s Theorem which states that whenever we have a class of algebras of the same type, we can first take all products, then all subalgebras and then all homomorphic images to get the variety generated by this class Homomorphisms When we think of homomorphisms, the first theorem that comes to mind is of course the homomorphism theorem. So in this subsection we will generalize this theorem for all universal algebras. Let us start with a recap of this theorem in group theory. Theorem 3.6 (Homomorphism theorem (for groups)). Let G and G be groups, and let f : G G be a surjective homomorphism. Then there exists an isomorphism h : G/ker(f) G defined by f = h g. Here g is the natural map from G to G/ker(f). In order to generalize this for all universal algebras we have to generalize the notion of the kernel and of the natural map. Let us start with the latter together with one of its properties. In the previous section we have defined congruences. When given a congruence on a universal algebra A we can define the natural map as follows. 20

21 Definition 3.7. Let A be an algebra and let θ be a congruence on A. The natural map ν θ : A A/θ is defined by ν θ = a/θ. It sends an element a to its congruence class by θ. Lemma 3.8. The natural map from an algebra to the quotient of the algebra by a congruence is a surjective homomorphism. Proof. Recall that a mapping α : A B is called a homomorphism if α(f A (a,..., a n )) = f B (α(a ),..., α(a n )). Given a congruence θ on A it is easy to see that the natural map ν θ as defined above is onto. So all we have to show is that ν θ is a homomorphism. To see this, suppose f is an n-ary operation in the type of A and B and a,..., a n A. Then we find ν θ (f A (a,..., a n )) = f A (a,..., a n )/θ = f A/θ (a /θ,..., a n /θ) = f A/θ (ν θ (a ),..., ν θ (a n )). Thus ν θ is a surjective homomorphism. Now that we have defined the natural map, let us define the kernel of a homomorphism. Definition 3.9. Let α : A B be a homomorphism. Then the kernel of α is defined by ker(α) = { a, b A 2 α(a) = α(b)} Lemma 3.0. Let α : A B be a homomorphism. Then ker(α) is a congruence on A. Proof. Suppose A and B are algebras of the same type and α : A B is a homomorphism. To see that ker(α) is a congruence we have to check the following properties: reflexivity, symmetry, transitivity and compatibility with the operations of A. In order to check this, suppose a, b, c A. (reflexivity) Trivially we have α(a) = α(a) thus a, a ker(α). (symmertry) If a, b ker(α) then α(a) = α(b) and of course α(b) = α(a) so we find b, a ker(α). (transitivity) Suppose a, b ker(α) and b, c ker(α). Then α(a) = α(b) and α(b) = α(c) thus α(a) = α(c) and therefore a, c ker(α). (compatibility) Now suppose f is an n-ary operation in the type of A and B. Also let us suppose that we have a i, b i ker(α) where i n. Then we have α(a i ) = α(b i ) for all i and therefore we have α(f A (a,..., a n )) = f B (α(a ),..., α(a n )) = f B (α(b ),..., α(b n )) = α(f A (b,..., b n )). Therefore f A (a,..., a n ), f A (b,..., b n ) ker(α). 2

22 Now we have obtained that ker(α) is indeed a congruence on A. Now that we have generalized the notions of natural map and kernel we can state the homomorphism theorem for universal algebras. Theorem 3. (Homomorphism Theorem). Suppose α : A B is an onto homomorphism. Then there is an isomorphism β : A/ker(α) B defined by α = β ν. Here ν is the natural map from A to A/ker(α). Proof. A proof of this theorem can be found in [, Theorem 6.2]. So when we take the natural homomorphism from an algebra A to A/ker(α) for any surjective homomorphism α from A to another algebra B, then we obtain an isomorphism between A/ker(α) and B. This is exactly what the homomorphism theorem tells us. Now let us define a homomorphic image of an algebra. Definition 3.2. Let α : A B be a homomorphism. Then α(a) is the homomorphic image of A by α. Note that when α is in addition onto, then α(a) = B and we have that B is a homomorphic image of A. We denote the set of all homomorphic images of an algebra A by H(A). Analogous we denote the set of all isomorphic images of an algebra A by I(A) Subalgebras The second ingredient for creating a variety is the notion of a subalgebra. In this section we give a definition of a subalgebra and we give an example of how this notion can be used. Definition 3.3. Let A and B be two algebras of type F. Then B is a subalgebra of A if B A and for every n-ary f F and b,..., b n B we have f B (b,..., b n ) = f A (b,..., b n ) i.e. every operation of B is the restriction of the corresponding operation of A. We can relate the definition of homomorphic image and subalgebra as follows. Lemma 3.4. Suppose α : A B is a one-to-one homomorphism. Then the homomorphic image of A is a subalgebra of B. Proof. By definition of α it follows that α(a) B. For the second property, suppose f is an n-ary operation of the type of A and B and suppose a,..., a n A. Then α(a ),..., α(a n ) α(a) and thus α(a ),..., α(a n ) B. Now we find and hence α(a) is a subalgebra of B. f α(a) (α(a ),..., α(a n )) = f B (α(a ),..., α(a n )) Let us denote the set of all subalgebras of an algebra A by S(A). 22

23 3.2.3 Products The last notion we need in order to start defining varieties is products. We will start this subsection by defining a (direct) product of two algebras together with the projection map. Then we will generalize this to the product of more (possibly infinitely many) algebras. At the end of this subsection we will show one of the properties of a product we are going to need later in this thesis. Definition 3.5. Let A and A 2 be two algebras of the same type. Define the (direct) product A A 2 to be the algebra whose set is A A 2, and such that for every n-ary operation and a i A, a i A 2 we have f A A 2 ( a, a,..., a n, a n ) = f A (a,..., a n ), f A 2 (a,..., a n). The product defined above gives rise to a map that projects an element to its i th coordinate. We call this map the projection of the algebra. Definition 3.6. For i {, 2} the mapping π i :A A 2 A i a, a 2 a i is called the projection map on the i th coordinate of the product. Lemma 3.7. The projection map is an onto homomorphism. Proof. The fact that π i is onto follows directly from the definition. To see that π i is a homomorphism, let a,..., a n A, a,..., a n A 2 and suppose f is an n-ary operation then for i = we find π (f A A 2 ( a, a,..., a n, a n )) = π ( f A (a,..., a n ), f A 2 (a,..., a n) ) = f A (a,..., a n ) = f A (π ( a, a ),..., π ( a n, a n )). We can do the same for every i = 2 and then obtain that the projection map is indeed a homomorphism. Now let us generalize the notion of product to a product of any family of algebras. Definition 3.8. Let {A i } i I be an indexed family of algebras of the same type. The (direct) product A = i I A i is an algebra with set i I A i and with the operations defined coordinate-wise: f A (a,..., a n )(i) = f A i (a (i),..., a n (i)). Here i I, f is n-ary and a,..., a n i I A i. And we also have projection maps as defined before: π j : A i A j i I 23

24 Given an indexed family of algebras of the same type K = {A i } i I, let us define the set of all products of algebras in K by P (K). We will need the next property when we give a characterization of subdirectly irreducible algebras in the last chapter. Lemma 3.9. For an indexed family of maps α i : A A i, i I, the following are equivalent: (i) The map α : A i I A i defined coordinate-wise by α(a)(i) = α i (a) is injective. (ii) i I ker(α i) =. Proof. Suppose a, a 2 A such that a a 2, then α is injective α(a ) α(a 2 ) i I : α(a )(i) α(a 2 )(i) i I : α i (a ) α i (a 2 ) i I : a, a 2 ker(a i ) i I ker(α i ) =. Thus α is injective if and only if i I ker(α i) = which is what we had to show Tarski s Theorem Now that we have established the definitions of homomorphic images, subalgebras and products, we can define the notion of varieties. In this subsection we will discuss different classes of algebras and some of their properties that will lead up to Tarski s Theorem. Definition A nonempty class K of algebras of the same type is called a variety if it is closed under homomorphic images, subalgebras and direct products. In the previous subsections we already have encountered the following notions in the special case where the class consists of only one algebra. We will now generalize this notion to classes consisting of more algebras. Definition 3.2. Let K be a class of algebras of the same type. Then A H(K) if and only if A is a homomorphic image of some member of K. A I(K) if and only if A is isomorphic to some member of K. A S(K) if and only if A is a subalgebra of some member of K. A P (K) if and only if A is a direct product of a nonempty family of algebras in K. From one class we can get bigger classes by applying H, I, S, or P to another class and in this way we can expand a class by first applying one operation and then another etc. In the following lemma we will find that some of the operations create bigger classes than others. Lemma For any class K of algebras of the same type we have 24

25 SH(K) HS(K) P H(K) HP (K) P S(K) SP (K) For the operations H, S and IP the idempotent law holds. Proof. Suppose K is a class of algebras of the same type. SH(K) HS(K) Let A an algebra in SH(K). Then there exists an algebra B H(K) such that A is a subalgebra of B. Since B H(K) there also exists an algebra C K such that B is a homomorphic image of C i.e. there exists an onto homomorphism α : C B. We have to prove that A HS(K) i.e. there exists an algebra D S(K) and an onto homomorphism β : D A and there exists an algebra E K such that D is a subalgebra of E. Claim: α (A) is a subalgebra of C. It is easy to see that α (A) = {c C α(c) A} C. So for the second condition, let f be an n-ary operation and c,..., c n α (A) then α(f α (A) (c,..., c n )) = f A (α(c ),..., α(c n )) (α is a homomorpism) = f B (α(c ),..., α(c n )) (A is a subalgebra of B) = α(f C (c,..., c n )). (α is a homomorpism) So we have proved the claim. Now since α(α (A)) = A and α is a homomorpism, we have established that A HS(K). P H(K) HP (K) Suppose A is an algebra in P H(K). Then there exist A i H(K) such that A = A i. Thus there are B i K such that there are α i : B i A i that are onto homomorphisms. Now let us define α : B A α(b)(i) α i (b(i)). Where B = B i. Now if α is an onto homomorphism, then A HP (K). Since for all i, α i is a surjective homomorphism, we find that α is also surjective and we have for b,..., b n B: α(f B (b,..., b n ))(i) = α i (f B i (b (i),..., b n (i))) = f A i (α i (b (i)),..., α i (b n (i))) = f A i (α(b )(i),..., α(b n )(i)) = f A (α(b ),..., α(b n ))(i). Thus α is an onto homomorphism. 25

26 P S(K) SP (K) Given an algebra A P S(K) then there exist A i S(K) such that A = A i. This yields that there exist B i K such that for every i, A i is a subalgebra of B i. Now let us define B to be B i then A SP (K) if A is a subalgebra of B. Since for every i we have A i is a subalgebra of B i thus for every i we have A i B i which implies A = A i B i = B and for f an n-ary operation we have f A (a,..., a n )(i) = f A i (a (i),..., a n (i)) = f B i (a (i),..., a n (i)) = f B (a,..., a n )(i). Here a,..., a n A. Therefore we have that A SP (K). H, S and IP are idempotent In order to see this, let A be an algebra. First suppose A HH(K) then there exists an algebra B H(K) and an onto homomorphism α : B A i.e. there exists an algebra C K and a homomorphism β : C B that is onto. Now let us show that α β is also an onto homomorphism, because then A H(K) and we find that for the operation H the idempotent law holds. Since α and β are both onto we have that α β is onto as well. Now suppose c,..., c n C then from the fact that both α and β are homomorphisms it follows that (α β)(f C (c,..., c n )) = α(β(f C (c,..., c n ))) = α(f B (β(c ),..., β(c n ))) = f A (α(β(c )),..., α(β(c n ))) = f A ((α β)(c ),..., (α β)(c n ). Thus α β is also a surjective homomorphism and A H(K). Now suppose A SS(K) then there exists an algebra B S(K) such that A is a subalgebra of B. Thus there is an algebra C K such that B is a subalgebra of C while A is a subalgebra of B. Now if we can show that A is also a subalgebra of C then we have established that SS is idempotent as well. Since A is a subalgebra of B and B is a subalgebra of C we have A B and B C thus by transitivity of we also have A C. Now suppose a,..., a n A then for f an n-ary operation we have f A (a,..., a n ) = f B (a,..., a n ) = f C (a,..., a n ) hence A is a subalgebra of C. To see that IP is also idempotent, suppose that A IP IP (K). Then there exists an algebra B P IP (K) that is isomorphic to A. Thus there exist B i IP (K) such that B = B i. Therefore there exist C i P (K) such that for every i, C i = Bi. Therefore there exist C ij K such that for every i we have C i = C ij. So we have A = B = B i = Ci = C ij. 26

27 Hence A = C ij. Since the product of a product is again a product we now have A IP (K). All algebras of the same type form a variety and since the intersection of any class of varieties is also a variety we know that there exists a smallest variety. Let us denote V (K) for the smallest variety that contains K. We will end this section with a theorem first proved by Alfred Tarski in 946 [3, pp 63-65]. Theorem 3.23 (Tarski). Let K be a class of algebras of the same type then we have V (K) = HSP (K). Proof. In order to prove this theorem we have to show two things: V (K) HSP (K) and V (K) HSP (K) where K is a class of algebras of the same type. Since V (K) is a variety we have that V is closed under H, S and P and thus HSP (K) V (K) follows directly from the definition. To see that HSP (K) is a variety (and thus V (K) HSP (K)) we have to show that HSP (K) is closed under homomorphic images, subalgebras and direct products. By Lemma 3.22 we know that the operation H is idempotent thus HHSP (K) = HSP (K) and we find that HSP (K) is closed under homomorphic images. From Lemma 3.22 we also know that the operation S is idempotent and SH(K) HS(K). Therefore we have SHSP (K) HSSP (K) = HSP (K). Thus HSP (K) is also closed under subalgebras. In order to see that HSP (K) is closed under products as well we use Lemma 3.22 again and find P HSP (K) HP SP (K) HSP P (K) HSIP IP (K) (P H(K) HP (K)) (P S(K) SP (K)) ( The identity map is an isomorphism) = HSIP (K) (IP is idempotent) HSHP (K) HHSP (K) = HSP (K). (any isomorphism is a homomorphism) (SH(K) HS(K)) Hence HSP (K) is closed under homomorphic images, subalgebras and products and thus HSP (K) is a variety. Now since HSP (K) V (K) and V (K) is the smallest variety that contains K we have established the equality V (K) = HSP (K). 3.3 Term algebras Now that we have defined varieties, we will use the rest of this chapter to characterize these classes as Garrett Birkhoff did in his paper in 935 [4, pp ]. He stated that 27

28 a class of algebras K is a variety if and only if K is an equational class. This theorem is known as one of Birkhoff s most famous theorems. In this paper we call it Birkhoff s Theorem. In order to fully understand what the theorem states we have to give a definition of an equational class. For this we first have to define terms in order to be able to define term algebras. We will use this to construct identities and then we can give a definition of an equational class. For the full proof of Birkhoff s Theorem we also need to understand the notion of the free algebra and some of its properties. For now, let us start by defining terms. Definition Given a type of algebras F. Let X be a set of variables. The set T (X) of terms of F over X is the smallest set such that (i) X {f f is a nullary operation in F} T (X) (ii) T (X) is closed under all the operations of F. For p in T (X) we say that p is n-ary if there are n or less variables occurring in p and we write p(x,..., x n ). The notion of terms is syntactic, just like the operations in a type: only the interpretation within an algebra has a meaning. So let us look at the meaning of a term in an algebra. Definition Given an algebra A and a term p(x,..., x n ) T (X) of the same type, let us define the map p A : A n A inductively: (i) If p(x,..., x n ) = x i for i n, then and p A is the i th projection map. p A (a,..., a n ) = a i (ii) If p = f(p (x,..., x n ),..., p k (x,..., x n )) for f a k-ary operation, then p A = f A (p A (x,..., x n ),..., p A k (x,..., x n )). p A is the interpretation of the term p in the algebra A. Now let us state some properties of terms. The proof can be found in [, Theorem 0.3]. Lemma For two algebras A and B of the same type and p T (X) n-ary, the following holds: (i) Let θ be a congruence on A and suppose a i, b i θ for i n. Then p A (a,..., a n ), p A (b,..., b n ) θ. 28

29 (ii) If α : A B is a homomorphism, then α(p A (a,..., a n )) = p B (α(a ),..., α(a n )). The set of terms also gives rise to an algebra: the term algebra. Definition Given a set X of variables and a type F of algebras, if T (X) then T(X) is the term algebra of F over X. The underlying set of this algebra is T (X) and the operations satisfy: f T(X) : T (X) n T (X) Here f F is n-ary and p,..., p n T (X). (p,..., p n ) f(p,..., p n ). Note that for T(X) to exist, either X or the set of nullary operations within the type has to be non-empty. Remark that T(X) is generated by the set X of variables i.e. T(X) is the smallest algebra such that the underlying set of T(X) contains X. The term algebra is one of the algebras for which the universal mapping property holds. Let us first define this property and finish this section with a prove of this statement. Definition Let K be a class of algebras of the same type F and let U(X) be an algebra generated by a set X. The algebra U(X) also has type F. If for every A K and for every map α : X A there is a homomorphism β : U(X) A x α(x) for all x X. Then we say that β extends α and that U(X) has the universal mapping property for K over X. Moreover, X is called the set of free generators of U(X), and U(X) is said to be freely generated by X. Note that for the universal mapping property to hold the map α need not be a homomorphism, α can be any map. Theorem For any type of algebras F and any set of variables X, the term algebra T(X) has the universal mapping property for the class of all algebras of type F over X. Proof. Suppose A is an algebra of type F and α : X A is a map. Now let us define another map β : T(X) A as follows: β(x) = α(x) for all x X; β(p(x,..., x n )) = p(β(x ),..., β(x n )); β(f T(X) (p (x,..., x n ),..., p k (x,..., x n ))) = f A (β(p (x,..., x n )),..., β(p k (x,..., x n ))). By definition β is a homomorphism. 29

30 3.4 Identities In some literature, identities are called equations or axioms, in this thesis however, we will stick to the term identities. In this section we will define the notion of identities and discuss some properties. We will end this section with a proof of one side of Birkhoff s Theorem. Definition Given a type of algebras F and a set X of variables. Let p, q T (X). An identity of F over X is an expression of the form p q. Let us define Id(X) to be the set of all identities of the type over X. Now we will define when an identity holds in an algebra. Definition 3.3. An identity p q holds in an algebra A if the interpretations of p and q are the same in A. Thus A = p(x,..., x n ) q(x,..., x n ) p A (a,..., a n ) = q A (a,..., a n ) for (a,..., a n ) A. When an identity holds in an algebra we write A = p q and we say that p q is true in A. A class K of algebras satisfies p q if each member of K satisfies p q, notation: K = p q. For a set of identities Σ we say that K satisfies Σ if K = p q for every identity p q Σ and we write K = Σ. Given a class of algebras K and a set X of variables, define We can characterize this as follows: Id K (X) = {p q Id(X) K = p q}. Lemma Given a class of algebras K of type F, a set of variables X and an identity of F over X, p q, then K = p q if and only if for every A K and for every homomorphism α : T(X) A we have α(p) = α(q). Proof. Suppose A K, p and q are n-ary terms and α : T(X) A is a homomorphism. Then by definition of = we have K = p q if and only if for every B K we have B = p q. Since A K this implies that A = p q. Now this tells us that for all 30

31 a,..., a n A we have p A (a,..., a n ) = q A (a,..., a n ). Since α is a homomorphism, by (ii) of Lemma 3.26 we have α(p T(X) (x,..., x n )) = p A (α(x ),..., α(x n )) = q A (α(x ),..., α(x n )) = α(q T(X) (x,..., x n )). Hence α(p) = α(q). Now for the other direction again suppose A K and p and q are n-ary terms. Furthermore assume that a,..., a n A. Now by Theorem 3.29 there exists a homomorphism α : T(X) A such that α(x i ) = a i for i n. Then we have p A (a,..., a n ) = p A (α(x ),..., α(x n )) = α(p T(X) (x,..., x n )) = α(p) = α(q) Hence K = p q. = α(q T(X) (x,..., x n )) = q A (α(x ),..., α(x n ) = q A (a,..., a n ). Now that we have defined identities (or equations as some might say), we can define equational classes: Definition Given a type of algebras F and a set of identities Σ of F. Define M(Σ) = {A A = Σ}. A class K of algebras is an equational class if there is a set of identities Σ such that K = M(Σ). In this case we say that K is defined or axiomatized by Σ. This is all we need to understand Birkhoff s Theorem which states that for every class of algebras K we have that K is an equational class if and only if K is a variety. We will prove one direction right now. In order to be able to prove the other direction we will need some more theory. We will prove it at the end of this chapter. Theorem 3.34 (Birkhoff part ). Every equational class is a variety. Proof. Let K be an equational class. Thus there exists a set of identities Σ such that M(Σ). A variety is a class of algebras which is closed under the operations H, S and P. So we have to show that M(Σ) is closed under these operations. 3

32 H Suppose A HM(Σ) i.e. there exists a B M(Σ) and a homomorphism α : B A which is onto. What we want is to show that A satisfies Σ. So let p q Σ and a,..., a n A. Since α is onto, we have that there exist b,..., b n B such that αb i = a i for i n. B M(Σ) thus p B (b,..., b n ) = q B (b,..., b n ) holds. So by Lemma 3.32 we have p A (a,..., a n ) = p A (αb,..., αb n ) = αp B (b,..., b n ) = αq B (b,..., b n ) = q A (αb,..., b n ) = q A (a,..., a n ). Thus A = Σ and therefore A M(Σ). S Suppose A SM(Σ) i.e. there exists a B M(Σ) such that A is a subalgebra of B. So there exists an embedding ι : A B defined by the restriction of the identity map to A. The identity map is a homomorphism thus again by Lemma 3.32 we have p A (a,..., a n ) = ι(p A (a,..., a n )) = p B (a,..., a n ) = q B (a,..., a n ) = ιq A ((a,..., a n )) = q A (a,..., a n ). Now A = Σ and thus A M(Σ). P Suppose A P M(Σ) i.e. there exist A i M(Σ) such that A = i I A i for some set I. Since for every i I the projection map π i : A A i is a surjective homomorphism we have p A (a,..., a n )(i) = π i (p A (a,..., a n )) = p A i (π i (a ),..., π i (a n )) = p A i (a (i),..., a n (i)) = q A i (a (i),..., a n (i)) = q A i (π i (a ),..., π i (a n )) = π i (q A (a,..., a n )) = q A (a,..., a n )(i). This holds for every i I so A = p q and A M(Σ). Thus M(Σ) is closed under homomorphic images, subalgebras and product and now we have found that every equational class is indeed a variety. 32