The algebraic theory of partial symmetries (or why groups aren’t enough) Mark V Lawson Heriot-Watt University and the Maxwell Institute for Mathematical Sciences Edinburgh, Scotland UK May 2012 1 1. Semigroups and monoids A semigroup is a set equipped with an associative binary operation. A monoid is a semigroup with an identity. Subsemigroups and submonoids are defined in the usual way. I shall now describe a couple of examples of semigroups. 2 2. The natural numbers with respect to addition A simple question: describe all submonoids and their structure. A special case of this problem is the following from recreational mathematics. You have an unlimited supply of 3 cent stamps and 5 cent stamps. By combining them, you can make up various denominations and so can send your letters to anywhere from Fiji to the Faroes. For example, 11 cents in value can be obtained by taking two 3 cent stamps and one 5 cent stamp. And so on. Now the question: what is the largest value you cannot make in this way? 3 This is called the theory of numerical semigroups. In particular, there are applications to algebraic geometry. 4 3. Combinatorics on words We now generalize the monoid of natural numbers to non-commutative monoids called free monoids. An alphabet is a finite set of symbols. A string is a finite sequence of symbols from an alphabet, with ε denoting the empty string. If A is an alphabet then A∗ is the set of all strings over A. 5 Given x, y ∈ A∗ the string xy is the concatenation of x and y. The set A∗ enjoys some additional properties. Observe that (xy)z = x(yz). Thus concatenation is associative. Also εx = x = xε. Thus the empty string is the identity for the operation of concatenation. The monoids of the form A∗ are called free monoids. If A consists of exactly one symbol then A∗ is just N with respect to addition. 6 Application 1: codes For example, {a, ba, b2} and {a2, ab, b} are both maximal prefix codes. The theory of codes is the theory of free submonoids of free monoids. 7 Application 2: syntactic monoids A language is a set of strings over an alphabet. Example: Welsh Alphabet all words from your favourite Welsh dictionary. Language W all grammatically correct sentences in Welsh constructed using this dictionary, such as iechyd da i chwi yn awr ac yn oesoedd From languages we can construct monoids. 8 Let L ⊆ A∗ be a language. Strings x, y ∈ A∗ belong to the same grammatical category with respect to the language L, and we write x σL y, if they occur in exactly the same contexts in the language. This means uxv ∈ L ⇔ uyv ∈ L. So, here, u v is the context in which the string x occurs. If L were a natural language, then grammatical categories would be things like nouns (or possibly special kinds of nouns: say, animate nouns, or feminine nouns etc), verbs, noun phrases etc. 9 If x ∈ A∗ we denote by [x]L the set of all strings in A∗ that are in the same grammatical category as x. Thus [x]L should be regarded as a grammatical category. The set of all grammatical categories of L is written A∗/σL We can multiply grammatical categories by defining [x]L[y]L = [xy]L. For example, the grammatical category nounphrase might be obtained by multiplying the grammatical categories determiner, adjective and noun together in that order. In this way, the set of all grammatical categories of L becomes a monoid called the syntactic monoid of the language L. 10 There are many other applications of free monoids: • Finite semigroup theory and its connections with finite state automata and regular languages. • Symbolic dynamics and through that to sundry mathematical fields. See the work of J. Rhodes (who exists) and M. Lothaire (who doesn’t). 11 This talk will now deal with a third example: the theory of inverse semigroups. 12 4. Inverse semigroups A semigroup S is said to be inverse when for each element s ∈ S there exists a unique element s−1 such that the following two equations hold s = ss−1s and s−1 = s−1ss−1. Two immediate examples. All groups. All meet semilattices. 13 Basic theory: (s−1)−1 = s and (st)−1 = t−1s−1 Elements of the form ss−1 and s−1s are idempotents; that is, equal to their square. Idempotents commute with each other. Groups are the inverse semigroups with exactly one idempotent (and so groups are degenerate inverse semigroups). Meet semilattices are the inverse semigroups in which every element is an idempotent. 14 Inverse subsemigroups are defined in the usual way as are homomorphisms and isomorphisms. Ky motivating example. Let X be a non-empty set. Denote by I(X) the set of all partial bijections of X. Then I(X) is an inverse monoid, called a symmetric inverse monoid. Theorem (Wagner-Preston) Every inverse semigroup is isomorphic to an inverse subsemigroup of a symmetric inverse monoid. Inverse semigroups are viewed as the theory of partial symmetries just as groups are the theory of symmetries. 15 The goal of the rest of this talk is to describe one interesting example of an inverse semigroup in depth. 16 5. The polycyclic inverse monoid P2 I shall define this inverse monoid as a set of partial symmetries of a particular fractal. Fractals have become part of the wall-paper of everyday life. The term was coined by Benoˆıt Mandelbrot in 1975, but his examples were drawn from developments in late 19th and early 20th century analysis. A formal definition is difficult, but self-similarity is an ingredient in what we regard as a fractal. 17 I shall take one of the simplest fractals: the Cantor set C. This is constructed by starting with the closed unit interval [0, 1] and succesively removing open middle-thirds ad infinitum. The self-similarity properties of this set are manifested by two maps p, q: C → C given by 2+x x p(x) = and q(x) = 3 3 and their respective ‘inverses’ p−1 and q −1. The polycyclic monoid on two generators is defined as the inverse submonoid of I(C) generated by p, q, p−1, q −1. 18 More concretely, the elements of the Cantor set may be identified with the right-infinite strings over a two-letter alphabet (a + b)ω . This leads to the following more convenient description of P2. Every non-zero element of P2 is of the form yx−1 where x and y are elements of the free monoid on {a, b}. The product of two elements yx−1 and vu−1 is zero unless x and v are prefix-comparable in which case yx−1 · vu−1 = ( yzu−1 if v = xz for some z y(uz)−1 if x = vz for some z The elements of P2 can be thought of as a combination of two operations on a pushdown stack: xy −1 means ‘pop y and push x’. 19 I shall now explain how P2 arises in algebraic linguistics. I shall define a family of languages called parentheses languages. Our alphabet An will consist of n matching pairs of brackets: x1, x ¯1, . . . , xn, x ¯n. Such languages arise naturally in many ways. For small n, it is more convenient to use different kinds of brackets. For example, when n = 2, we could use, say, (, ), [, ]. The language Ln is the set of all correct bracketing sequences. This can be made precise, but I shall make do with an example: ( [ ( ) ] {, } [ ], ) ( ) 20 Parenthesis languages are of a type known as context-free (CF), which are amongst the most important classes of languages, and play a distinguished role in their theory. Theorem [Chomsky-Sch¨ utzenberger] Every CF language is an alphabet image of the intersection of a language of type Ln for some n and a local language. This tells us that parenthesis languages are the templates from which all CF languages can be constructed. Local languages are very simple kinds of regular languages essentially arising from finite directed graphs. 21 We are interested in the syntactic monoid of L2. This can be computed using the minimal automaton of L2. This machine is an infinite binary tree (with an additional dump state). The syntactic monoid of L2 is just the transition monoid of this machine. In particular, x1x ¯1 = 1 and x2x ¯2 = 1 and x ¯2x1 = 0 and x ¯1x2, where 0 comes from the dump state. Theorem The syntactic monoid of L2 is P2. 22 6. A completion of P2 The way I have defined P2 it comes equipped with an action on (a + b)ω . I am now going to write down three equations, the Cantor relations, that summarize the important properties of P2 and its action. (1). 1 = p−1p = q −1q (2). 0 = p−1q = q −1p (3). 1 = pp−1 + qq −1 where 1 means the identity function defined on C. I claim that where you see these equations, you see self-similarity analogous to that of the Cantor set. 23 Examples 1. Let N be the set of natural numbers and E and O the sets of even and odd numbers, respectively. Let p: N → E be the doubling map, and let q: N → O be the doubling-and-add-one map. Cantor’s relations hold and tell us that ℵ0 + ℵ0 = ℵ0. 24 2. Let R be a unital ring that contains elements p, p−1, q, q −1 satisfying Cantor’s relations. Let M2(R) denote the set of all 2 × 2 matrices over R. We expect M2(R) to be ‘bigger than’ R. 25 Define v= p−1 ! q −1 and h = p q ! . Observe that vht = 1 0 0 1 ! and htv = (1) where we have utilized all the Cantor relations. In what follows, I identify (x) and x. 26 Define Φ: M2(R) → R by A 7→ htAv. Φ is an injective homomorphism Define Ψ: R → M2(R) by r 7→ vrht. Ψ is an injective homomorphism Φ and Ψ are mutually inverse. Conclusion: M2(R) is isomorphic to R. 27 3. Banach-Tarski paradox. A closed ball B in R3 can be partitioned into two pieces B1 and B2 in such a way that each piece Bi is piecewise congruent to B. Subsets X and Y of R3 are said to be piecewise congruent if X can be partitioned into a finite number of pieces that can be moved in space using only translations and rotations and then glued back together to yield Y . 28 We may glue suitable elements of P2 together. Here is an example. Consider α = a2a−1 + ab(ba)−1 + bb−2. We have that • a2a−1: aAω → a2Aω is given by aw 7→ a2w. • ab(ba)−1: baAω → abAω is given by baw 7→ abw. • bb−1: b2Aω → bAω is given by b2w 7→ bw. The set {a, ba, b2} is a maximal prefix code as is the set {a2, ab, b}. We may draw a picture of what α is doing. 29 A final theorem By extending the previous example, the polycyclic inverse monoid P2 may be completed to an inverse monoid C2 called the Cuntz inverse monoid. This inverse monoid is closely related to a C ∗algebra, the Cuntz C ∗-algebra O2. The group of units of C2 is Thompson’s group V, an infinite finitely presented simple group. 30 Envoi Its soon, no sense, that faddoms the herts o men, And by my sangs the rouch auld Scots I ken Een herts that hae nae Scots’ll dirl richt thro As nocht else could — for here’s a language rings Wi datchie sesames, and names for nameless things. Gairmscoile, by Hugh MacDiarmid Tapadh leat! 31

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