Lectures on Universal Algebra Matt Valeriote McMaster University November 8, 1999 1 Algebras In this ¯rst section we will consider some common features of familiar alge- braic structures such as groups, rings, lattices, and boolean algebras to arrive at a de¯nition of a general algebraic structure. Recall that a group G consists of a nonempty set G, along with a binary ¡1 operation ¢ : G ! G, a unary operation : G ! G, and a constant 1G such that ² x ¢ (y ¢ z) = (x ¢ y) ¢ z for all x, y, z 2 G, ¡1 ¡1 ² x ¢ x = 1G and x ¢ x = 1G for all x 2 G, ² 1G ¢ x = x and x ¢ 1G = x for all x 2 G. A group is abelian if it additionally satis¯es: x ¢ y = y ¢ x for all x, y 2 G. A ring R is a nonempty set R along with binary operations +, ¢, a unary operation ¡, and constants 0R and 1R which satisfy ² R, along with +, ¡, and 0R is an abelian group. ² x ¢ (y ¢ z) = (x ¢ y) ¢ z for all x, y, z 2 G. ² x ¢ (y + z) = (x ¢ y) + (x ¢ z) and (y + z) ¢ x = (y ¢ x) + (z ¢ x) for all x, y, z 2 G. 1 ² 1G ¢ x = x and x ¢ 1G = x for all x 2 G. Lattices are algebras of a di®erent nature, they are essentially an algebraic encoding of partially ordered sets which have the property that any pair of elements of the ordered set has a least upper bound and a greatest lower bound. A lattice L consists of a nonempty set L, equipped with two binary operations ^ and _ which satisfy: ² x ^ x = x and x _ x = x for all x 2 L, ² x ^ y = y ^ x and x _ y = y _ x for all x, y 2 L, ² x ^ (y _ x) = x and x _ (y ^ x) = x for all x, y 2 L. It follows from the above equations, that if we de¯ne the relation · on L by: x · y if and only if x = x^y then · is a partial order on L. Furthermore, for x, y 2 L, the element x ^ y is the greatest lower bound of x and y, and x _ y is the least upper bound of x and y with respect to the order ·. Conversely, if hP; ·i is a partially ordered set, with the property that each pair from P has a greatest lower bound and least upper bound, then by de¯ning x ^ y and x _ y to be the greatest lower bound and least upper bound of x and y respectively, P , equipped with these operations is a lattice. Sometimes lattices have a smallest element and a largest element with respect to its partial order. When this is the case, we use the constant symbols 0 and 1 to denote these elements. Such a lattice is called a bounded lattice and satis¯es the equations: x ^ 1 = x and x _ 0 = x for all x 2 L. A lattice is distributive if it satis¯es the distributive laws: x ^ (y _ z) = (x ^ y) _ (x ^ z) and x _ (y ^ z) = (x _ y) ^ (x _ z): One last important class of algebras to mention is the class of boolean algebras. A boolean algebra is an algebra of the form hB; ^; _;0 ; 0; 1i where hB; ^; _; 0; 1i is a bounded distributive lattice and 0 is a unary operation such that ² (x0)0 = x for all x 2 B. 2 ² (x ^ y)0 = x0 _ y0 for all x, y 2 B. ² (x _ y)0 = x0 ^ y0 for all x, y 2 B. ² x0 _ x = 1 and x0 ^ x = 0 for all x 2 B. The set of all subsets of a set X, with the operations of intersection, union and complementation forms a boolean algebra called a ¯eld of sets. It turns out that every boolean algebra can be embedded into some ¯eld of sets. De¯nition 1 An algebra A is a pair hA; F i, with A a nonempty set, called the universe of A, and F = hfi : i 2 Ii, a sequence of ¯nitary operations on A. The operations in F are called the basic operations of A and the set I is called the index set of A. The type of A is the function ¿ : I ! N, where ¿(i) is equal to the arity of the function fi. (The arity of an operation f on A is n, if and only if the domain of f is An.) Two algebras are said to be similar if and only if they have the same type. We often call the elements of the index set I of an algebra A the operation symbols of A, and for f 2 I, the basic operation of A corresponding to f is denoted by f A. Following this convention then, the index set of the ring of integers is f¢; +; ¡; 0; 1g, while the operation of multiplication is denoted by ¢Z. If the context is clear, we usually dispense with the superscript. Note that all of the examples presented earlier are algebras in the above sense. Those familiar with groups and rings will undoubtably be familiar with the concepts of subgroups and subrings, as well as with homomorphisms, iso- morphisms, and direct products. All of these things have natural extensions to the class of all algebras. De¯nition 2 A subuniverse of the algebra A is a subset (possibly empty) of A which is closed under all of the basic operations of A. B is a subalgebra of A if B is similar to A, B is a nonempty subuniverse of A, and for each f in the index of A, the operation f B is the restriction to B of the operation f A. Let Sub (A) be the set of all subuniverses of the algebra A. This set is naturally ordered by inclusion. It turns out that this ordered set is a lattice. 3 Proposition 3 Let A be an algebra. 1. The intersection of a set S of subuniverses of A is a subuniverse of A, and is the greatest lower bound of S. 2. Let A be an algebra. Then Sub (A), ordered by inclusion is a lattice. From this proposition, we see that intersection is the meet operation of the subuniverse lattice. It also follows that for X ⊆ A, the intersection of the set of all subuniverses of A which contain X is a subuniverse of X. Clearly, this subuniverse is the smallest one which contains X, and is denoted by Sg A(X). If B = Sg A(X), then X is called a generating set for B. If B is generated by a ¯nite set, then B is said to be a ¯nitely generated subuniverse. The algebra A is called ¯nitely generated if A is a ¯nitely generated subuniverse of A. With a little e®ort, it can be shown that the join of two subuniverses U and V of A is equal to Sg A(U [ V ). A homomorphism from the group G to the group H can be described as a map from G to H which is compatible with the operations of the groups. Lifting this to arbitrary algebras yields the following de¯nition: De¯nition 4 Let A and B be similar algebras, of type ¿ : I ! N.A function h : A ! B is a homomorphism from A to B if h is compatible with the basic operations of A and B, i.e., for all f 2 I, if ¿(f) = n, then for all A B a1; : : : ; an 2 A, h(f (a1; : : : ; an)) = f (h(a1); : : : ; h(an)). If h : A ! B is a surjective homomorphism from A to B, then B is called a homomorphic image of A. An isomorphism between two similar algebras A and B is a homomorphism which is also injective and surjective. In this case, the algebras A and B are said to be isomorphic. For h a homomorphism between a pair of groups or rings, we are able to associate a subalgebra, called the kernel of h, by setting the kernel to be the set of elements in the domain of h which are mapped to the identity ele- ment of the range. Besides being a subalgebra, the kernel has other special properties. Since a general algebra doesn't usually come equipped with an identity element, then we can't naturally associate a subalgebra with a ho- momorphism. Instead, we de¯ne the kernel of a homomorphism h : A ! B, denoted by ker(h), to be the following equivalence relation: ker(h) = f(a; b) 2 A £ A : h(a) = h(b)g: 4 The fact that h is compatible with the basic operations of A and B carries over to the kernel. Namely, if (ai; bi) 2 ker(h) for i · n and f is an n-ary basic operation of A, then (f(a1; : : : ; an); f(b1; : : : ; bn)) 2 ker(h). As with groups and rings, the kernel can be used to detect when a homo- morphism is injective. Namely, Proposition 5 Let h : A ! B be a homomorphism from A to B. Then h is injective if and only if ker(h) = f(a; a): a 2 Ag. Note that for h a group homomorphism, the kernel of h in the above sense is equal to the equivalence relation induced by the partitioning of the domain by the cosets of the kernel in the usual sense.