Dimensional functions over partially ordered sets - - PowerPoint PPT Presentation

Intro Dimensional functions over posets Application I Application II

Dimensional functions over partially ordered sets

V.N.Remeslennikov, E. Frenkel May 30, 2013

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Intro Dimensional functions over posets Application I Application II

Plan The notion of a dimensional function over a partially ordered set was introduced by V. N. Remeslennikov in 2012. Outline of the talk: Part I. Definition and fundamental results on dimensional functions, (based on the paper of V. N. Remeslennikov and

• A. N. Rybalov “Dimensional functions over posets”);

Part II. 1st application: Definition of dimension for arbitrary algebraic systems; Part III. 2nd application: Definition of dimension for regular subsets of free groups (L. Frenkel and V. N. Remeslennikov “Dimensional functions for regular subsets of free groups”, work in progress).

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Intro Dimensional functions over posets Application I Application II

Partially ordered sets Definition A partial order is a binary relation ≤ over a set M such that ∀a ∈ M a ≤ a (reflexivity); ∀a, b ∈ M a ≤ b and b ≤ a implies a = b (antisymmetry); ∀a, b, c ∈ M a ≤ b and b ≤ c implies a ≤ c (transitivity). Definition A set M with a partial order is called a partially ordered set (poset).

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Intro Dimensional functions over posets Application I Application II

Linearly ordered abelian groups Definition A set A equipped with addition + and a linear order ≤ is called linearly ordered abelian group if

• 1. A, + is an abelian group;
• 2. A, ≤ is a linearly ordered set;
• 3. ∀a, b, c ∈ A a ≤ b implies a + c ≤ b + c.

Definition The semigroup A+ of all nonnegative elements of A is defined by A+ = {a ∈ A | 0 ≤ a}.

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Intro Dimensional functions over posets Application I Application II

A−dimensional functions Let M be a poset and A be a linearly ordered abelian group. Definition The function d : M → A+ is called A-dimensional over M if ∀x, y ∈ M if x < y in M, then d(x) < d(y) in A.

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Intro Dimensional functions over posets Application I Application II

Dense dimensional functions Definition An A-dimensional function d : M → A+ is called dense if for all x, y ∈ M such that d(x) < d(y) there exist elements x′ and y ′ in M satisfying d(x) ≤ d(x′), d(y ′) ≤ d(y) and x′ < y ′.

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Intro Dimensional functions over posets Application I Application II

Strongly dense dimensional functions Definition An A−dimensional function d : M → A+ is called strongly dense if for every x, y ∈ M such that d(x) < d(y) there exist elements x′, y ′ satisfying d(x) = d(x′), d(y ′) = d(y) and x′ < y ′.

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Intro Dimensional functions over posets Application I Application II

Flows The A− dimensional function d : M → A+ defines an equivalence relation ∼d on M by m1 ∼d m2 ↔ d(m1) = d(m2). Let [m1] ≤d [m2] ↔ d(m1) ≤ d(m2). Then the linearly ordered set M/ ∼d is a homomorphic image of M. Definition The set M/ ∼d is called a d−flow. The order type of a d−flow is denoted by πd(M).

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Intro Dimensional functions over posets Application I Application II

Equivalence of dimensional functions Definition Let M be a poset and suppose d1, d2 are dimensional functions

• ver M with values in some linearly ordered abelian groups. Then

d1 ∼ d2 if the order types πd1(M) and πd2(M) are isomorphic. Fact A poset M may have non-equivalent dimensional functions.

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Intro Dimensional functions over posets Application I Application II

Example of non-equivalent dimensional functions Example Let L1 = {[0, 1], 2, 3, [4, 5]} and L2 = {[6, 7], 8, [9, 10]}, with the natural order on them, and suppose that all elements of L1 and L2 are non-comparable. Let M = L1 ∪ L2. Then M admits non-equivalent dimensional functions. Define d1 : M → R as follows: let d1 shift all elements of L2 to the left by 1, and let it fix L1. In this case, πd1(M) = {[0, 1], 2, 3, [4, 6], 7, [8, 9]}. Define d2 : M → R as follows: let d2 map L2 into {[−4, −3], −2, [−1, 0]}, and let it fix L1. In this case, πd2(M) = {[−4, −3], −2, [−1, 1], 2, 3, [4, 5]}. Clearly, d1, d2 are dimensional functions, but πd1(M) and πd2(M) are non-isomorphic. Therefore, d1 is not equivalent to d2.

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Intro Dimensional functions over posets Application I Application II

The case of finite posets Proposition Let M be a finite poset. Then, up to the equivalence relation defined above, there exist only one Z-dimensional function over M. In particular, there exist only one strongly dense Z-dimensional function over M in this equivalence class.

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Intro Dimensional functions over posets Application I Application II

The category of posets with dimensional functions Let L be a language (or a signature) defining the category of posets with dimensional functions. Then L is the disjoint union of three languages: LM = {≤M}, where ≤M is a binary predicate; LA = {+, −, ≤A, 0}, where + is a binary predicate (addition), − is a unary predicate (inversion), and ≤A is a binary predicate of

• rder, 0 is a constant symbol;

L3 = {δM, δA, d} consists of two unary predicates that distinguish sets M and A and a binary predicate d corresponding to the graph

• f the dimensional function.

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Intro Dimensional functions over posets Application I Application II

The category of posets with dimensional functions We define the category K of posets with dimensional functions

• ver L using the following 4 groups of axioms:

Disjoint union of underlying sets M′ = M ⊔ A, where M is the underlying set of the predicate δM, and A is the underlying set

• f the predicate δA.

Axioms of partial order on M. Axioms of abelian linearly ordered group A. Axioms of dimensional functions d : M → A+.

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Intro Dimensional functions over posets Application I Application II

Existence of dimensional functions Theorem 1. For every poset M there exist a linearly ordered abelian group A and a dimensional function d : M → A+.

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Intro Dimensional functions over posets Application I Application II

Discrete linearly ordered abelian groups Definition A linearly ordered abelian group A is called discrete if A+ has minimal nonzero element (denoted by 1A). Theorem If for a poset M and discrete linearly ordered group A there exists a dimensional function d : M → A+, then there exists a dense dimensional function d∗ : M → A+.

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for direct products Theorem Let di : Mi → A be a dimensional function for a poset Mi, i = 1, 2. Then the function d : M1 × M2 → A such that ∀m1 ∈ M1 ∀m2 ∈ M2 d((m1, m2)) = d1(m1) + d2(m2), is a dimensional function for the direct product M1 × M2.

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Intro Dimensional functions over posets Application I Application II

Ordinal dimensional functions Definition An A−dimensional function on a poset M is called ordinal, if the

• rder type πd(M) is a well ordered set.

A poset M is called a set of ordinal type, if there exists a dense

• rdinal dimensional function for M.

Definition A poset M is called Artinian if any chain a1 > a2 > . . . in M is finite (i.e. satisfies DCC).

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Intro Dimensional functions over posets Application I Application II

Ordinal dimensional functions Theorem 2.

• 1. If a poset M has an ordinal dimensional function, then it is an

Artinian poset.

• 2. For an Artinian poset there exists a unique (up to

equivalence) dense ordinal A−dimensional function.

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Intro Dimensional functions over posets Application I Application II

Lattice dimensional functions A poset L is a lattice, if

• 1. for any two elements x, y ∈ L, the set {a, b} has the greatest

lower bound (x ∧ y), and

• 2. for any two elements x, y ∈ L, the set {a, b} has the least

upper bound (x ∨ y).

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Intro Dimensional functions over posets Application I Application II

Lattice dimensional functions Uuniversal algebra point of view: L is an algebraic system with two binary operations ∧ and ∨ satisfying universal identities: (L1) Laws of idempotency: ∀a a ∧ a = a, a ∨ a = a. (L2) Commutativity laws: ∀a, b a ∧ b = b ∧ a, a ∨ b = b ∨ a. (L3) Associativity laws: ∀a, b, c (a ∧ b) ∧ c = a ∧ (b ∧ c), (a ∨ b) ∨ c = a ∨ (b ∨ c). (L4) Absorption laws: ∀a, b a ∧ (a ∨ b) = a, a ∨ (a ∧ b) = a. Using these operations, one can define a partial order: a ≤ b ↔ a ∧ b = a.

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Intro Dimensional functions over posets Application I Application II

Lattice dimensional functions Definition Let a poset M be a lattice, and let A be a linearly ordered abelian

• group. A function d : M → A+ is called a lattice A-dimensional

function if

1

d is A-dimensional function.

2

∀x, y ∈ M d(x ∨ y) + d(x ∧ y) = d(x) + d(y).

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Intro Dimensional functions over posets Application I Application II

Modular lattices Definition A lattice L is called modular, if ∀x∀y∀z ∈ L z ≤ x → x ∧ (y ∨ z) = (x ∧ y) ∨ z. For a modular lattice L with zero one can define a height function h : L → N: h(a) is the length of the longest maximal chain in the interval [0, a], if it exists, and h(a) = ∞, otherwise. Definition A lattice L is called a finite length lattice if h(a) < ∞ for all a ∈ L.

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Intro Dimensional functions over posets Application I Application II

Modular lattices Jordan-Goelder theorem In a finite length modular lattice every two maximal chains in [0, a] have the same length and for all a, b ∈ L the following equality holds h(a) + h(b) = h(a ∧ b) + h(a ∨ b). From this condition it follows that for a finite length modular lattice there exists a lattice Z-dimensional function: the length function. The notion of lattice dimensional function allows us to transfer the notion of height to a wide class of modular lattices, preserving main properties of this notion.

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Intro Dimensional functions over posets Application I Application II

Lattice dimensional functions Theorem. Let Lh be a class of lattices L such that there exists linearly

• rdered abelian group A and lattice A−dimensional function for L.

Then the class Lh is axiomatizable in the language {L, ≤, d}.

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Intro Dimensional functions over posets Application I Application II

Lattice dimensional functions: existence theorem Theorem 3. For the following classes of modular lattices there exist lattice dimensional functions: locally finite modular lattices, distributive lattices, boolean lattices.

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Intro Dimensional functions over posets Application I Application II

Applications Application I.

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Intro Dimensional functions over posets Application I Application II

Krull dimension for commutative Noetherian rings Definition Let k be a field, R be a commutative Noetherian k−algebra. Then the Krull dimension dimk(R) is the upper bound of the set of lengths of all prime ideals in R. Here the length of a prime ideal P in R is the upper bound of all integers m such that there exists a chain P0 P1 P1 . . . Pm = P of prime ideals of R. The set of prime ideals of R is called the spectrum (Spec(R)). The set Spec(R) admits a natural partial order.

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Intro Dimensional functions over posets Application I Application II

Krull dimension for commutative Noetherian rings From our point of view: dimk(R) is Z−dimensional function d : Spec(R) → Z. Let kn be the n−dimensional affine space, AlgSn be a partially

• rdered set of all algebraic subsets of X over k, and Γ(X) be the

coordinate ring. Then the function dim(X) = dimk(Γ(X)) is a Z− dimensional function for AlgSn. (One can find the definitions we use below in a series of papers on Universal Algebraic Geometry by Daniyarova, Miasnikov, R.)

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Intro Dimensional functions over posets Application I Application II

Krull dimension for commutative Noetherian rings Let S be an algebraic system for some language L, let X be a nonempty algebraic set over S, and Γ(X) be its coordinate system. Define Spec(Γ(X)) to be the set of simple congruences for Γ(X); then there is a natural partial order on this set. By Theorem 1 for the poset Spec(Γ(X)) there exists a discretely

• rdered abelian group A and a dimensional function

d : Spec(Γ(X)) → A+. Definition The element d(X) is called the d−dimension of the algebraic set X.

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Intro Dimensional functions over posets Application I Application II

Equationally Noetherian algebraic systems Theorem Let S be an equationally Noetherian algebraic system. Then for the category of algebraic sets AlgSn there exists a unique (up to equivalence) ordinal dense dimensional function (d−dimension).

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Intro Dimensional functions over posets Application I Application II

Algebro-geometric ordinals for equationally Noetherian algebraic systems The ordinal type of the set Sn does not depend on the choice of A and is a well-defined ordinal αn(S). Define AGD(S) as AGD(S) = (α1, α2, . . .). We shall call it algebro-geometric dimension of S.

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Intro Dimensional functions over posets Application I Application II

Algebro-geometric ordinals for equationally Noetherian algebraic systems Question. Does there exist a hyperbolic group G such that AGD(G) is a sequence of infinite ordinals? General question. Let S be an arbitrary equationally Noetherian algebraic system. What is AGD(S)? Example+Question. Let S be a free non-abelian Lie algebra over the field k. Then α1(S) = ω. (Daniyarova, R.) Is it true that α2(S) = 2ω?

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Intro Dimensional functions over posets Application I Application II

Applications Application II.

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for regular subsets of a free group Let F = F(X) be a free group of finite rank. One can use a no-return random walk Ws (s ∈ (0, 1]) on the Cayley graph C(F, X) as a random generator of elements of F. We start at the identity element 1 and either do nothing with probability s (and return value 1 as the output of our random word generator),

• r move to one of the 2m adjacent vertices with equal probabilities

(1 − s)/2m. If we are at a vertex v = 1, we either stop at v with probability s (and return the value v as the output), or move, with probability

1−s 2m−1, to one of the 2m − 1 adjacent vertices lying away

from 1, thus producing a new freely reduced word vx±1

i

. It is easy to see that the probability µs(w) for our process to terminate at a word w is given by the formula µs(w) = s(1 − s)|w| 2m · (2m − 1)|w|−1 for w = 1 and µs(1) = s.

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for regular subsets of a free group Let R be a subset of the free group F and Sk = { w ∈ F | |w| = k } be the sphere of radius k in F. The ratio fk(R) = |R ∩ Sk| |Sk| is the frequency of elements from R among the words of length k in F. For R ⊆ F its measure µs(R) is defined by µs(R) =

w∈R µs(w).

Recalculating µs(R) in terms of s, one gets µs(R) = s

∞

• k=0

fk(1 − s)k, and the series on the right hand side is convergent for all s ∈ (0, 1). The collection of distributions {µs} can be encoded in a single function µ(R) : s ∈ (0, 1) → µs(R) ∈ R.

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for regular subsets of a free group Put s = 0 and obtain a non-stopping random walk on the Cayley graph C(F, X). In this case the probability λ(w) that the walker reaches an element w ∈ F in |w| steps equals λ(w) = 1 2m(2m − 1)|w|−1 , if w = 1, and λ(1) = 1.

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for regular subsets of a free group For a subset R of F we define the limit measure µ0(R) (the Cesaro density): µ0(R) = limn→∞ 1 n(f1 + . . . + fn). The Cesaro density and λ−measure for regular subsets of F allow us to introduce the following notion of dimension on a poset of regular subsets of F (cf. asymptotic classification of subsets of F).

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Intro Dimensional functions over posets Application I Application II

Dimensional functions for regular subsets of a free group Let R be a regular subset of F(X) and let A = Q × Q with left lexicographical order. For any element of the class R = {R is regular in F(X)} we define a map d : R → A+ by d(R) = (µ0(R), λ(R0), ) where the negligible set R0 can be effectively constructed by R. Theorem The map d : R → A+ is an A−dimensional function.

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