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An Introduction to Algebra Tom Bourne University of St Andrews PIMS, 26th January 2015 Tom Bourne An Introduction to Algebra Semigroups, Monoids and Groups Definition A set X equipped with a binary operation ◦ : X × X → X is a semigroup if: x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ; Tom Bourne An Introduction to Algebra Semigroups, Monoids and Groups Definition A set X equipped with a binary operation ◦ : X × X → X is a monoid if: x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ; ∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ; Tom Bourne An Introduction to Algebra Semigroups, Monoids and Groups Definition A set X equipped with a binary operation ◦ : X × X → X is a group if: x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ; ∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ; ∀x ∈ X ∃y ∈ X such that x ◦ y = 1X = y ◦ x. Tom Bourne An Introduction to Algebra Semigroups, Monoids and Groups Definition A set X equipped with a binary operation ◦ : X × X → X is a group if: x ◦ (y ◦ z) = (x ◦ y ) ◦ z ∀x, y , z ∈ X ; ∃1X ∈ X such that 1X ◦ x = x = x ◦ 1X ∀x ∈ X ; ∀x ∈ X ∃y ∈ X such that x ◦ y = 1X = y ◦ x. We have the following inclusions: groups ⊂ monoids ⊂ semigroups. Between us we work with the above structures in a variety of different settings, and, generally, we do not consider more complicated examples such as rings and fields. Tom Bourne An Introduction to Algebra Equivalence Relations and Congruences Definition A binary relation ∼ on a set X is an equivalence relation if and only if for all x, y , z ∈ X we have: x ∼ x (reflexive); if x ∼ y then y ∼ x (symmetric); if x ∼ y and y ∼ z then x ∼ z (transitive). Definition A congruence is an equivalence relation on an algebraic structure that is compatible with the multiplication defined on that structure. For example, if (G , ◦) is a group then a congruence on G is an equivalence relation ≡ satisfying g1 ≡ h1 and g2 ≡ h2 ⇒ g1 ◦ g2 ≡ h1 ◦ h2 ∀g1 , g2 , h1 , h2 ∈ G . Tom Bourne An Introduction to Algebra Green’s Relations Green’s relations, written as L, R, J , H and D, are semigroup equivalences defined in terms of principal ideals. We define the relations over the monoid S 1 , which is the semigroup S with an identity adjoined if necessary. For elements a, b in a semigroup S, we define Green’s relations by: a L b ⇔ S 1 a = S 1 b; a R b ⇔ aS 1 = bS 1 ; a J b ⇔ S 1 aS 1 = S 1 bS 1 ; a H b ⇔ a L b and a R b; a D b ⇔ ∃c ∈ S such that a L c and c R b. Unfortunately, Green’s relations tell us nothing useful when dealing with groups. Tom Bourne An Introduction to Algebra Subgroups, Cosets and Normality Definition A subset H of a group G is a subgroup if H is a group in its own right with binary operation inherited from G . We write H ≤ G . Let G be a group and H ≤ G . For g ∈ G , the left coset of H in G with respect to g is the set gH = {gh | h ∈ H}. We define right cosets similarly. Definition A subgroup N of G is normal if and only if gN = Ng ∀g ∈ G ; that is, the sets of left and right cosets coincide. We write N / G . Tom Bourne An Introduction to Algebra Simplicity and Quotient Groups Definition A group G is simple if its only normal subgroups are the trivial group and G itself. Let G be a group and N / G . Let G /N denote the set of cosets of N in G . Define a binary operation on G /N by (gN)(hN) = (gh)N. Definition The set G /N together with the binary operation defined above is the quotient or factor group of N in G . Essentially, we have placed an equivalence relation on G and partitioned the set into its equivalence classes to form G /N. Tom Bourne An Introduction to Algebra Free Objects I Let A be an alphabet; that is, a non-empty finite set of letters. A word (over A) is a finite sequence of letters from A. The set of all non-empty words equipped with the operation of word concatenation is the free semigroup on A, denoted A+ . If we also include the empty word of length zero then we have the free monoid on A, and we denote this by A∗ . For the free group, we take an alphabet A and the corresponding set of ‘inverse’ symbols A−1 . Let B = A ∪ A−1 , and define a word to be a finite sequence of letters from B. If a letter and its inverse appear adjacent in a word then we identify them with the empty word to form an equivalent reduced word. The set of all reduced words (including the empty word) equipped with the operation of word concatenation is the free group on A. Tom Bourne An Introduction to Algebra Free Objects II In general, we refer to an arbitrary structure as free if it is isomorphic to the corresponding free structure. The free semigroup/monoid forms the basis of formal language theory as we define a language over an alphabet A to be a subset of the free semigroup/monoid on A. For example, if A = {a, b} then the set of all words starting with an a and ending with a b, denoted aA∗ b, and the set {an b n | n ≥ 0} are both languages over A. Tom Bourne An Introduction to Algebra Graphs Definition A graph is an ordered pair G = (V , E ), where V is a set of vertices and E is a set of unordered pairs from V × V . The elements of E are edges. [Too lazy to code and too forgetful to remember: draw graph now.] If we insist that the elements of E are ordered; that is, the edge (1, 2) is not the same as (2, 1), then the graph is directed. We use (directed) graphs to represent other structures... Tom Bourne An Introduction to Algebra Automata Definition The common components of an automaton, denoted A = (S, A, δ, . . .), are: a finite set of states S; an input alphabet A; a transition function δ. The addition of other features, such as start/final states, output alphabets, and stacks, together with a defined domain and range for δ allow the machine to process different information. For example, finite state automata correspond to regular languages whereas pushdown automata correspond to context-free languages. The most well-known type of automaton is a Turing machine. Tom Bourne An Introduction to Algebra Thompson’s Groups Thompson’s groups F , T and V were introduced in the 1960s as a potential counter-example to the von Neumann conjecture. All three groups are infinite but finitely presented, and satisfy the inclusions F ⊆ T ⊆ V . The group F can be realised in terms of operations on ordered rooted binary trees, which are special types of undirected graphs. It can also be found in topological settings. Tom Bourne An Introduction to Algebra Computational Algebra and GAP Computational algebra, or symbolic computation, is the area of study dedicated to the development of algorithms and software, primarily for manipulating mathematical objects and expressions. Such software includes Maple and Mathematica. Another system is GAP (Groups, Algorithms and Programming), which the University of St Andrews is a development centre for. It is a system for computational discrete algebra with particular emphasis on group theory, though numerous packages can be added-on to the main software. One such package is ‘Semigroups’, which is co-authored by a selection of PhD students. Tom Bourne An Introduction to Algebra