Intro Dimensional functions over posets Application I Application II

Dimensional functions over partially ordered sets

V.N.Remeslennikov, E. Frenkel

May 30, 2013

1 / 38 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). 2 / 38 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).

3 / 38 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}.

4 / 38 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.

5 / 38 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 ′.

6 / 38 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 ′.

7 / 38 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).

8 / 38 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 over 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.

9 / 38 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. 10 / 38 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.

11 / 38 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 order, 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 of the dimensional function.

12 / 38 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 over 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 of the predicate δA.

Axioms of partial order on M.

Axioms of abelian linearly ordered group A.

Axioms of dimensional functions d : M → A+.

13 / 38 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+.

14 / 38 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+.

15 / 38 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.

16 / 38 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 order type πd (M) is a well ordered set. A poset M is called a set of ordinal type, if there exists a dense ordinal dimensional function for M.

Definition

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

17 / 38 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.

18 / 38 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).

19 / 38 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: ∀aa ∧ a = a, a ∨ a = a.

(L2) Commutativity laws: ∀a, ba ∧ 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.

20 / 38 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).

21 / 38 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.

22 / 38 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.

23 / 38 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 ordered abelian group A and lattice A−dimensional function for L. Then the class Lh is axiomatizable in the language {L, ≤, d}.

24 / 38 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.

25 / 38 Intro Dimensional functions over posets Application I Application II Applications

Application I.

26 / 38 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.

27 / 38 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 ordered 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.)

28 / 38 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 ordered 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 .

29 / 38 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).

30 / 38 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.

31 / 38 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?

32 / 38 Intro Dimensional functions over posets Application I Application II Applications

Application II.

33 / 38 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), or 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 1−s probability 2m−1 , to one of the 2m − 1 adjacent vertices lying away ±1 from 1, thus producing a new freely reduced word vxi . It is easy to see that the probability µs (w) for our process to terminate at a word w is given by the formula s(1 − s)|w| µs (w)= for w = 1 2m (2m − 1)|w|−1 and µs (1) = s. 34 / 38 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 |R ∩ Sk | fk (R)= |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)= Pw∈R µs (w). Recalculating µs (R) in terms of s, one gets ∞ k µs (R)= s X fk (1 − s) , k=0 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

s µ(R) : s ∈ (0, 1) → µ (R) ∈ R. 35 / 38 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 1 λ(w)= , if w = 1, and λ(1) = 1. 2m(2m − 1)|w|−1

36 / 38 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): 1 µ (R)= limn (f + . . . + fn). 0 →∞ n 1 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 ).

37 / 38 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.

38 / 38