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Dimensional Functions Over Partially Ordered Sets

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Dimensional Functions Over Partially Ordered Sets 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).
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