INTRODUCTION TO ABSTRACT ALGEBRA G. JANELIDZE Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch 7701, Cape Town, South Africa Last updated on 20 September 2013 This is an elementary course aiming to introduce very first concepts of abstract algebra for third year students interested in pure mathematics. It is divided into the following (two chapters and) sections: I. First algebraic and related structures 1. Algebraic operations..........................................2 2. Magmas and unitary magmas.....................................3 3. Semigroups....................................................4 4. Monoids.......................................................5 5. Closure operators.............................................5 6. Equivalence relations.........................................7 7. Order relations...............................................8 8. Categories....................................................9 9. Isomorphism..................................................11 10. Initial and terminal objects.................................12 11. Algebras, homomorphisms, isomorphisms........................13 12. Subalgebras..................................................15 13. Products.....................................................16 14. Quotient algebras............................................16 15. Canonical factorization of homomorphisms.....................17 16. Classical algebraic structures...............................18 17. Quotient groups, rings, and modules..........................20 II. Lattices, semirings, number systems, and foundation of linear algebra 1. Commutativity................................................25 2. Semilattices.................................................26 3. Lattices.....................................................29 4. Distributivity and complements...............................31 5. Complete lattices............................................33 6. Boolean algebras.............................................34 7. Semirings and semimodules....................................37 8. The semiring ℕ of natural numbers............................38 9. Number systems, ℤ, ℚ, ℝ, and ℂ as rings, and their modules...40 10. Pointed categories...........................................46 11. Products and coproducts......................................47 12. Direct sums..................................................51 13. Free algebras and free semimodules...........................52 14. Vector spaces................................................55 We shall refer to these sections as follows: say, “Section 10” means “Section 10 in this chapter”, while “Section I.10” means “Section 10 in Chapter I”, when we refer to it in Chapter II. I am grateful to Dr. Amartya Goswami for a number of misprint corrections. 1 I. FIRST ALGEBRAIC AND RELATED STRUCTURES 1. Algebraic operations For a natural number n and a set A, the set of all maps from {1,…,n} to A will be denoted by An; this includes the case n = 0, where {1,…,n} becomes the empty set and therefore An becomes a one-element set. For n = 1, the set An will be identified with A, and for n = 2, 3,…, the elements of An will be written as n-member sequences (a1,…,an) of elements in A. Definition 1.1. For n = 0, 1, 2,…, an n-ary operation on a set A is a map from An to A. We will also use special terms for small n’s: 0-ary nullary 1-ary unary 2-ary binary 3-ary ternary Remarks and Conventions 1.2. (a) Since to give a nullary operation on A is the same as to pick up an element in A, we will simply identify them; accordingly the nullary operations are sometimes called constants. (b) If is an n-ary operation with n = 2, 3,…, we will write (a1,…,an) instead of ((a1,…,an)). Furthermore, for n = 2, instead of (a1,a2) we usually write a1a2, or simply a1a2 – especially when we think of as a kind of multiplication. Binary operations play an especially important role. When the ground set A has a small number of elements, it is convenient to define binary operations on it with tables, such as: a b a a a b a b where A = {a,b} and aa = ab = ba = a, bb = b, or a b a b b b b b where A = {a,b} again, but now aa = ab = ba = bb = b. That is, when A has n elements, the table contains n 1 rows and n 1 columns with the following entries in their cells: the first cell is blank, and the rest of the first row lists the elements of A in any order; the rest of the first column lists the elements of A (preferably) in the same order; 2 for i and j both greater than 1, the cell on the intersection of i-row and j-column has the element ab in it – where a stays in the i-th cell of the first column and b stays in the j-th cell of the first row. Definition 1.3. Let be a binary operation on a set A, and let us write (a,b) = ab, as above. The operation is said to be (a) associative, if a(bc) = (ab)c for all a, b, c in A; (b) commutative, if ab = ba for all a, b in A; (c) idempotent, if aa = a for all a in A. 2. Magmas and unitary magmas Definition 2.1. A magma1 is a pair (M,m), where M is a set and m a binary operation on M. Convention 2.2. Whenever only one magma (M,m) is considered, we will always write m(a,b) = ab and often write just M instead of (M,m). Definition 2.3. An element e in a magma M is said to be: (a) an idempotent, if ee = e; (b) a left identity (or a left unit), if ea = a for every a in M; (c) a right identity (or a right unit), if ae = a for every a in M; (d) an identity2 (or a unit), if it is a left and a right identity at the same time, i.e. if ea = a = ae for every a in M. Theorem 2.3. In an arbitrary magma: (a) every left identity and every right identity is an idempotent; (b) if e is a left identity and e' is a right identity, then e = e'. Proof. (a) is trivial. (b): e = ee' = e'. Remark 2.4. For an arbitrary set M one can define a binary operation on it by ab = b for all a and b in M. In the resulting magma, every element is a left identity. And of course one can do the same with the right identities by putting put ab = a. On the other hand Theorem 2.3(b) tells us that the existence of at least one left identity and at least one right identity in the same magma immediately implies that all of them are equal, yielding a unique identity. Definition 2.5. A unitary magma is a triple (M,e,m), where (M,m) be a magma and e its identity. 1 In old literature magmas were sometimes called groupoids; this has been changed in order to avoid confusions with groupoids in category theory. 2 As follows from 2.3(b) below (see also Remark 2.4), we could also say “the identity”. 3 Convention 2.6. Whenever only one unitary magma (M,e,m) is considered, we will (still use Convention 2.2 and) write 1 instead of e. Remark 2.7. As we see from Remark 2.4, it is a triviality that every unitary magma has exactly one left identity, exactly one right identity, and exactly one identity (all the same). 3. Semigroups Definition 3.1. A magma (M,m) is called a semigroup if m is associative. Theorem 3.2. If (M,m) is a semigroup, then there exists a unique sequence of operations m1, m2, … on M such that: (a) mn is an n-ary operation (n = 1, 2,…); (b) m1(a) = a for every a in M; (c) mp(a1,…,ap)mq(b1,…,bq) = mp+q(a1,…,ap,b1,…,bq) for all p = 1, 2,…; q = 1, 2,…; and a1,…, ap, b1,…, bq in M. Proof. Existence. Let us define m1, m2, … inductively by (b) for m1 and by mn+1(a1,…,an1) = mn(a1,…,an)an1 for m2, m3, …, i.e. for n = 1, 2,…. After this we will prove (c) by induction in q as follows: for q = 1 we have mp(a1,…,ap)mq(b1) = mp(a1,…,ap)b1 = mp1(a1,…,ap,b) for each p; for q > 1, using the inductive assumption and associativity, we then obtain mp(a1,…,ap)mq(b1,…,bq) = mp(a1,…,ap)(mq1(b1,…,bq1)bq) = (mp(a1,…,ap)mq1(b1,…,bq1))bq = mpq1(a1,…,ap,b1,…,bq1)bq = mp+q(a1,…,ap,b1,…,bq), as desired. Uniqueness. From our assumptions on the sequence m1, m2, … we obtain mp+1(a1,…,ap1) = mp(a1,…,ap)m1(ap1) = mp(a1,…,ap)ap1 and so our inductive definition was a consequence of those assumptions. According to Convention 2.2, it is also convenient to avoid writing the letter m for the operations introduced in Theorem 3.2. That is, one writes n n mn(a1,…,an) = a1…an = ai = i=1ai (3.1) i=1 and, in this notation, the formula given in 3.2(c) becomes 4 (a1…ap)(ap1…apq) = a1…apq, (3.2) or, equivalently p pq pq (i=1ai)(i =p1ai) = i =1ai, (3.3) n Furthermore, for a1 = … = an = a, one writes a instead of a1…an, and (3.2) becomes apaq = apq. (3.4) 4. Monoids Definition 4.1. A monoid is a unitary magma (M,e,m), in which m is associative. In other words, a monoid is a unitary magma (M,e,m), in which (M,m), is a semigroup. Theorem 3.2 reformulates for monoids as follows: Theorem 4.2. If (M,e,m) is a monoid, then there exists a unique sequence of operations m0, m1, … on M such that: (a) mn is an n-ary operation (n = 0, 1,…); (b) m1(a) = a for every a in M; (c) mp(a1,…,ap)mq(b1,…,bq) = mp+q(a1,…,ap,b1,…,bq) for all p = 0, 1,…; q = 0, 1,…; and a1,…, ap, b1,…, bq in M. Proof. As follows from Theorem 3.2, all we need is to take care of m0. That is, given a sequence m1, m2, … as in Theorem 3.2, we need to prove that there exists a unique element m0 in M satisfying m0m0 = m0, mp(a1,…,ap)m0 = mp(a1,…,ap), and m0mq(b1,…,bq) = mq(b1,…,bq) for all p = 1, 2,…; q = 1, 2,…; and a1,…, ap, b1,…, bq in M.