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 m 1 ∼ d m 2 ↔ d ( m 1 ) = d ( m 2 ). Let [ m 1 ] ≤ d [ m 2 ] ↔ d ( m 1 ) ≤ d ( m 2 ). 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 π d ( M ). 8 / 38
Intro Dimensional functions over posets Application I Application II Equivalence of dimensional functions Definition Let M be a poset and suppose d 1 , d 2 are dimensional functions over M with values in some linearly ordered abelian groups. Then d 1 ∼ d 2 if the order types π d 1 ( M ) and π d 2 ( 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 L 1 = { [0 , 1] , 2 , 3 , [4 , 5] } and L 2 = { [6 , 7] , 8 , [9 , 10] } , with the natural order on them, and suppose that all elements of L 1 and L 2 are non-comparable. Let M = L 1 ∪ L 2 . Then M admits non-equivalent dimensional functions. Define d 1 : M → R as follows: let d 1 shift all elements of L 2 to the left by 1, and let it fix L 1 . In this case, π d 1 ( M ) = { [0 , 1] , 2 , 3 , [4 , 6] , 7 , [8 , 9] } . Define d 2 : M → R as follows: let d 2 map L 2 into { [ − 4 , − 3] , − 2 , [ − 1 , 0] } , and let it fix L 1 . In this case, π d 2 ( M ) = { [ − 4 , − 3] , − 2 , [ − 1 , 1] , 2 , 3 , [4 , 5] } . Clearly, d 1 , d 2 are dimensional functions, but π d 1 ( M ) and π d 2 ( M ) are non-isomorphic. Therefore, d 1 is not equivalent to d 2 . 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: L M = {≤ M } , where ≤ M is a binary predicate; L A = { + , − , ≤ 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; L 3 = { δ M , δ A , 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 δ M , and A is the underlying set of the predicate δ A . 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 1 A ). 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 d i : M i → A be a dimensional function for a poset M i , i = 1 , 2. Then the function d : M 1 × M 2 → A such that ∀ m 1 ∈ M 1 ∀ m 2 ∈ M 2 d (( m 1 , m 2 )) = d 1 ( m 1 ) + d 2 ( m 2 ) , is a dimensional function for the direct product M 1 × M 2 . 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 π d ( 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 a 1 > a 2 > . . . 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: ∀ 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 . 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 d is A -dimensional function. 1 ∀ x , y ∈ M d ( x ∨ y ) + d ( x ∧ y ) = d ( x ) + d ( y ). 2 21 / 38
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