Monoids. Definition: A binary operation on a set M is a function : M M M. Examples:

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1 Monoids Definition: A binary operation on a set M is a function : M M M. If : M M M, we say that is well defined on M or equivalently, that M is closed under the operation. Examples:

2 Definition: A monoid is a pair (M, ) where M is a set M satisfying: (A0) M is closed under : : M M M. (A1) identity element e M: e a = a e = a, a M. (A2) Associativity: Note: Due to (A2), we can denote a b c = a (b c) = (a b) c. Examples: a (b c) = (a b) c, a, b, c

3 Definition: The elements of a subset A M are called generators of the monoid (M, ) if any element y M with the exception of the identity element can be obtained from them by successively applying the operation : y M \ {e}, a 1,..., a n A, y = a 1 a 2... a n. Examples: Definition: A set of generators A M is called free if any element y M can be written in terms of the elements of A in a unique way, i.e. there are no two different ways of writing m using and elements in A: a 1,..., a n, b 1,..., b m A, y = a 1... a n = b 1... b m m = n, a i = b i, i.

4 Example: Definition: A monoid with a free set of generators is called a free monoid. Definition: (N, +) is the free monoid generated by 1. Other examples: Definition: A commutative (or Abelian) monoid is a monoid (M, ) where also satisfies: Commutativity: a, b M, a b = b a.

5 Addition on N: 1) Consider the set N given by the Constructive Definition. Definition: We define + : N N N by A + B = A B for any sets A and B such that A B =. Note: (N, +) thus defined is a monoid: (A1) + has identity element 0 = : A + = A = A, A finite set. (A2) + is associative because the union of sets is: A (B C) = (A B) C = A + ( B + C ) = ( A + B ) + C for A, B, C disjoint. Exercise: Prove that the monoid (N, +) is commutative: (A3) a, b N, a + b = b + a.

6 Multiplication on N: 1) Consider the set N given by the Constructive Definition. Definition: We define : N N N by A B = A B. Note: (N, ) thus defined is a monoid: (A1) has identity element 1 = { } : A { } = A { } = A, A finite set. (A2) + is associative: A (B C) = (A B) C. Moreover, the new operation satisfies: (D) is distributive with respect to +, i.e. a, b, c N, a(b + c) = ab + ac. Similarly, (b + c)a = ba + ca. Indeed, for all finite sets A, B, C, A (B C) = A B A C, (B C) A = B A C A,

7 Finite monoids Definition:Let (M, ) be a monoid with M = finite set. The Cayley table of (M, ) is the table whose rows/columns correspond to the elements of M, and whose entry on the row a and column b is a b. * b a a b Examples: Z 2 = {[0], [1]}, where [0] = { all even numbers } and [1] = { all odd numbers }. + [0] [1] 1. The Cayley table for the monoid (Z 2, +): [0] [0] [1] [1] [1] [0] + [0] [1] 2. The Cayley table for the monoid (Z 2, ): [0] [0] [0] [1] [0] [1] 3. Z 5 = {[0], [1], [2], [3], [4]}, where [0] = {... 10, 5, 0, 5, 10, 15,...} = { all multiples of 5}, [1] = {... 9, 4, 1, 6, 11, 16,...} = = { numbers whose division by 5 yields remainder 1}, [2] = {... 8, 3, 2, 7, 12, 17,...} = = { numbers whose division by 5 yields remainder 2}... For convenience, when it is clear that we work modulo a certain number, we can write the equivalence classes without the square brackets, i.e. we ll write 2 and mean [2].

8 The Cayley table for the monoid (Z 5, +): = = 5 0 ( mod 5), = = 6 1 ( mod 5), because = = 7 2 ( mod 5), = 8 3 ( mod 5) is a generator for (Z 5, +) because n = }{{} n times (Z 5, +) is not freely generated by 1 because. e.g., 2 can be written in two different ways using 1 and +: 2 = = }{{} 7 times 4. The Cayley table for the monoid (Z 5, ): because (Z 5, +) is generated by {0, 2} because 2 3 = 6 1 ( mod 5), 2 4 = 8 3 ( mod 5), 3 3 = 9 4 ( mod 5), 3 4 = 12 2 ( mod 5) 4 4 = 16 1 ( mod 5). 0 = 0 in Z 5, 1 = identity elem. in Z 5, 2 = 2 in Z 5, 3 = in Z 5 4 = 2 2 in Z 5. (Z 5, +) is not freely generated by {0, 2} because. e.g., 1 = so 2 = (Z 5, +) is generated by {0, 3} because 0 = 0 in Z 5, 1 = identity elem. in Z 5, 2 = in Z 5, 3 = 3 in Z 5 4 = 3 3 in Z 5.

9 (Z 5, +) is not freely generated by {0, 3} because. e.g., 1 = so 3 = (Z 5, +) is not generated by {0, 4} because 4 4 = 1 in Z 5 so only 0, 1 and 4 can be written using {0, 4} and. Other examples of monoids 1. (M m n (N), +)= the monoid of m n matrices with entries in N, where + is the matrix addition. (M m n (N), +) is generated by {E ij ; i {1,..., m} and j {1,..., n}}, where E ij is the matrix having the entry on the i-th row and j-th column equal to 1, all other entries equal to For every set X, the set of functions {f : X X} together with, the composition of functions, forms a monoid. 3. The set of functions {f : R R; f continuous } together with, the composition of functions, forms a monoid. Monoid Homomorphisms Previously we claimed that (N, +) is the free monoid generated by one element, because for any n N nonzero, n = 1 } + {{ }. n times

10 On the other hand, consider the set M = {e, a, aa, aaa,...} of all words made out of the letter a, with the concatenation operation : a...a }{{} m times a...a }{{} n times = a...a }{{} m+n times Here e denotes the empty word, which is the identity element for. Then (M, ) has equal claims to the title of free monoid generated by one element. Although (N, +) and (M, ) are two different monoids, it is clear that one can be obtained from the other by relabeling: Thus relabel 0 by e, then n by a...a }{{} and n times + by, and (N, +) has become (M, ). If we could write the Cayley table of (N, +),. + n m m + n then relabel, we d get * a...a }{{} n times a...a }{{} m times a...a }{{} m+n times which is the Cayley table of (M, ). In fact we have constructed a map f : N M which sends the monoid operation + to the operation.

11 Definition: Let (X, ) and (Y, ) be two monoids with identity elements e X and e Y, respectively. A monoid homomorphism f : (X, ) (Y, ) is a function f : X Y such that f(e X ) = e Y for all m, n X. and f(m n) = f(m) f(n), Definition: A monoid homomorphism f : (X, ) (Y, ) which is bijective is called monoid isomorphism. Notation: In this case we say that (X, ) and (Y, ) are isomorphic, and write (X, ) = (Y, ). Example: 1. (N, +) = (M, ). We say that (N, +), the free monoid with one generator, uniquely defined up to an isomorphism (i.e. by a relabeling of its elements and its operation.) 2. f : (N, +) (N, ) given by f(x) = e x is a monoid homomorphism, injective but not surjective. 3. Let (M, +) be a monoid and let be another binary operation on M. For each element a M, define f : M M by f(x) = a x. Prove that f : (M, +) (M, +) is a monoid homomorphism iff satisfies the distributivity condition for all x, y M. a (x + y) = a x + a y, 4. Consider the map f : Z 3 Z 3 given by f(n) = n + 1. Define a new operation on Z 3 such that f : (Z 3, ) (Z 3, ) is a monoid isomorphism.

12 References An elementary introduction to sets, relations, functions: duentsch/archive/methprimer1.pdf The same, as taught by a philosophy professor: Some more advanced texts on Set Theory, the Axiom of Choice and the Set of Natural Numbers: theory.pdf holmes/holmes/head.pdf Further reading. Pick your own:

2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.

2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a. Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms