Quotient Set Research Articles

Difference sets are not multiplicatively closed

Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp. The famous sum-product problem of Erdős and Szemeredi asks the following. Let $A$ be a set of $n$ real numbers. Define the _sumset_ $A+A$ of $A$ to be the set $\{x+y:x,y\in A\}$ and the _product set_ $A.A$ to be the set $\{xy:x,y\in A\}$. Must at least one of $A+A$ and $A.A$ have size $n^{2-o(1)}$? Since neither set can have size greater than $n^2$, this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was $n^{4/3}$, up to logarithmic factors [3], though the exponent has now been very slightly improved [2]. More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if $A$ is a subset of a finite field ${\mathbb F}_p$ and $p^{\delta} < |A| < p^{1-\delta}$, then one of $A+A$ and $A.A$ must have size at least $n^{1+\epsilon}$ for some $\epsilon$ that depends on $\delta$ only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that $\mathbb F_p$ does not. This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if $A$ has a large difference set $D=A-A$, it is perfectly possible for $D+D$ to be much bigger than $D$. Nevertheless, when $A$ is a set of real numbers, it still implies, as the main result of the paper shows, that the product set $D.D$ has size at least $|D|^{1+c}$, where $c>0$ is an absolute constant. The same is true of the quotient set $D/D$. The paper also proves a result of a similar kind in the finite field $\mathbb F_p$ provided that $D$ is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup $H$ of $\mathbb F_p$ that is not too large is not the difference set of any subset of $\mathbb F_p$. (The precise assumption required is that $H$ has order at most $p^{4/5-\epsilon}$.) [1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343) [2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771) [3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040) [Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)

Quotient algebraic structures on the set of fuzzy numbers

A. M. Bica has constructed in [6] two isomorphic Abelian groups, defined on quotient sets of the set of those unimodal fuzzy numbers which have strictly monotone and continuous sides. In this paper, we extend the results of above mentioned paper, to a larger class of fuzzy numbers, by adding the flat fuzzy numbers. Furthermore, we add the topological structure and we characterize the constructed quotient groups, by using the set of the continuous functions with bounded variation, defined on $[0,1]$.

On the Duality between Existence and Information

Recent developments in pure mathematics (category theory) and in mathematical logic (partition logic) have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets (mis-specified as the logic of "propositions") and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction (a pair of elements in different blocks) of a partition, and that leads to a whole stream of dualities or analogies -- including the development of new logical foundations for information theory parallel to Boole's development of logical finite probability theory. After outlining these dual concepts in mathematical terms, we turn to a more metaphysical speculation about two dual notions of reality, a fully definite notion using Boolean logic and appropriate for classical physics, and the other objectively indefinite notion using partition logic which turns out to be appropriate for quantum mechanics. The existence-information duality is used to intuitively illustrate these two dual notions of reality. The elucidation of the objectively indefinite notion of reality leads to the "killer application" of the existence-information duality, namely the interpretation of quantum mechanics.

Four Quotient Set Gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.

Group Structures on Quotient Sets in Classification Problems

The relationship between the properties of difunctionality and ternary relations is analyzed as applied to information factorization problems. Conditions are found under which arbitrary n-ary relations induce multialgebraic systems with a common carrier in the form of a Cartesian cube. Multigroup axiomatics is considered. Conditions of existence of groups over equivalence classes are formulated and proved. Examples of producing multigroups and cases when a common carrier is absent are given.

Classification of finite-dimensional irreducible modules over -algebras

AbstractFinite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

Partitions and Objective Indefiniteness in Quantum Mechanics

Classical physics and physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties all the way down, and the notion of an objectively indefinite type of reality. The problem of interpreting mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual to one another and which are developed in two dual mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is follow the math by showing how the mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome toy or pedagogical model of quantum mechanics over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret mechanics.

Bounded sets of sheaves on Kähler manifolds

Abstract We show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

THE THRESHOLD DECISION MAKING EFFECTUATED BY THE ENUMERATING PREFERENCE FUNCTION

Based on the leximin and leximax preferences, we consider two threshold preference relations on the set X of alternatives, each of which is characterized by an n-dimensional vector (n ≥ 2) with integer components varying between 1 and m(m ≥ 2). We determine explicitly in terms of binomial coefficients the unique utility function for each of the two relations, which in addition maps X onto the natural 'interval' [Formula: see text], where [Formula: see text] is the quotient set of X with respect to the indifference relation I on X induced by the threshold preference. This permits us to evaluate all equivalence classes and indifference classes of the threshold order on X, present an algorithm of ordering the monotone representatives of indifference classes, and restore the indifference class of an alternative via its ordinal number with respect to the threshold preference order.

Atomistic Subsemirings of the Lattice of Subspaces of an Algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1 A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.

On algebras with a generalized implication

Abstract We introduce the notion of gi-algebra as a generalization of dual BCK-algebra, and define the notions of strong, commutative and transitive gi-algebra, and then we show that an interval ↑l = {a ∈ P | l ≤ a} in a strong and commutative gi-algebra P is a lattice. Also, we define a congruence relation ∼D on a transitive gi-algebra P and show that the quotient set P/∼D is a gi-algebra and a dual BCK-algebra.

Various improvements to text fingerprinting

Let s = s 1 . . s n be a text (or sequence) on a finite alphabet Σ of size σ . A fingerprint in s is the set of distinct characters appearing in one of its substrings. The problem considered here is to compute the set F of all fingerprints of all substrings of s in order to answer efficiently certain questions on this set. A substring s i . . s j is a maximal location for a fingerprint f ∈ F (denoted by 〈 i , j 〉 ) if the alphabet of s i . . s j is f and s i − 1 , s j + 1 , if defined, are not in f . The set of maximal locations in s is L (it is easy to see that | L | ⩽ n σ ). Two maximal locations 〈 i , j 〉 and 〈 k , l 〉 such that s i . . s j = s k . . s l are named copies , and the quotient set of L according to the copy relation is denoted by L C . We first present new exact efficient algorithms and data structures for the following three problems: (1) to compute F ; (2) given f as a set of distinct characters in Σ , to answer if f represents a fingerprint in F ; (3) given f , to find all maximal locations of f in s . As well as in papers concerning succinct data structures, in the paper all space complexities are counted in bits. Problem 1 is solved either in O ( n + | L C | log σ ) worst-case time (in this paper all logarithms are intended as base two logarithms) using O ( ( n + | L C | + | F | log σ ) log n ) bits of space, or in O ( n + | L | log σ ) randomized expected time using O ( ( n + | F | log σ ) log n ) bits of space. Problem 2 is solved either in O ( | f | ) expected time if only O ( | f | log n ) bits of working space for queries is allowed, or in worst-case O ( | f | / ϵ ) time if a working space of O ( σ ϵ log n ) bits is allowed (with ϵ a constant satisfying 0 < ϵ < 1 ). These algorithms use a data structure that occupies | F | ( 2 log σ + log 2 e ) ( 1 + o ( 1 ) ) bits. Problem 3 is solved with the same time complexity as Problem 2, but with the addition of an occ term to each of the complexities, where occ is the number of maximal locations corresponding to the given fingerprint. Our solution of this last problem requires a data structure that occupies O ( ( n + | L C | ) log n ) bits of memory. In the second part of our paper we present a novel Monte Carlo approximate construction approach. Problem 1 is thus solved in O ( n + | L | ) expected time using O ( | F | log n ) bits of space but the algorithm is incorrect with an extremely small probability that can be bounded in advance.

Algebraic properties for some permutation statistics

In this article, we study some quotient sets on permutations built from peaks, valleys, double rises and double descents. One part is dedicated to the enumeration of the cosets using the bijection of Francon-Viennot which is a bijection between permutations and the so-called Laguerre histories. Then we study the algebraic properties of these quotient sets. After having shown that some of them give rise to quotient algebras of $\mathbf{FQSym}$, we prove that they are also free.

Non-Linear Images of &lt;em&gt;μ&lt;/em&gt;-Shadings, Shadings in ℝ&lt;sup&gt;2&lt;/sup&gt;, and Quotient Sets of &lt;em&gt;μ&lt;/em&gt;-Shadings

In the paper we prove that some natural modifications (for instance, images under some functions, Cartesian products, quotient sets) of certain types of \(\mu\)-shadings (or shadings), are other examples of \(\mu\)-shadings (or shadings). The studies of shadings and \(\mu\)-shadings were initiated by R. Mabry in 1990. Our work is a continuation of his and K. Neu's research in this field. In particular, we solve one problem posed by R. Mabry.

Quotient sets and density recurrent sets

Quotient sets and density recurrent sets

State-Dependent Implication and Equivalence in Quantum Logic

Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Luders state). It is shown that two events and give the same Luders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Luders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed.

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the fibers are of unbounded size.

On pallets for Fox colorings of spatial graphs

We introduce the notion of pallets of quandles and define coloring invariants for spatial graphs which give a generalization of Fox colorings studied in Ishii and Yasuhara (1997) [4]. All pallets for dihedral quandles are obtained from the quotient sets of the universal pallets under a certain equivalence relation. We study the quotient sets and classify their elements.

Quotient Sets and Diophantine Equations

Quotient sets 핌/핌 = {u/u′ : u, u′ ∊ 핌} have been considered several times before in the Monthly. We consider more general quotient sets 핌/핍 and we apply our results to certain simultaneous Diophantine equations with side constraints.

A Hasse diagram for rational toral ranks

Let $X$ be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of $X$ obtained by almost free toral actions, ${\mathcal T}_0(X)=\{[Y_i] \}$, induced by an equivalence relation based on rational toral ranks, we order as $[Y_i] 0$. It presents a variation of almost free toral actions on $X$. We consider about the Hasse diagram ${\mathcal H}(X)$ of the poset ${\mathcal T}_0(X)$, which makes a based graph $G{\mathcal H}(X)$, with some examples. Finally we will try to regard $G{\mathcal H}(X)$ as the 1-skeleton of a finite CW complex ${\mathcal T}(X)$ with base point $X_{\Q}$.