Semiring of Sets
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Sketch Notes — Rings and Fields
Sketch Notes — Rings and Fields Neil Donaldson Fall 2018 Text • An Introduction to Abstract Algebra, John Fraleigh, 7th Ed 2003, Adison–Wesley (optional). Brief reminder of groups You should be familiar with the majority of what follows though there is a lot of time to remind yourself of the harder material! Try to complete the proofs of any results yourself. Definition. A binary structure (G, ·) is a set G together with a function · : G × G ! G. We say that G is closed under · and typically write · as juxtaposition.1 A semigroup is an associative binary structure: 8x, y, z 2 G, x(yz) = (xy)z A monoid is a semigroup with an identity element: 9e 2 G such that 8x 2 G, ex = xe = x A group is a monoid in which every element has an inverse: 8x 2 G, 9x−1 2 G such that xx−1 = x−1x = e A binary structure is commutative if 8x, y 2 G, xy = yx. A group with a commutative structure is termed abelian. A subgroup2 is a non-empty subset H ⊆ G which remains a group under the same binary operation. We write H ≤ G. Lemma. H is a subgroup of G if and only if it is a non-empty subset of G closed under multiplication and inverses in G. Standard examples of groups: sets of numbers under addition (Z, Q, R, nZ, etc.), matrix groups. Standard families: cyclic, symmetric, alternating, dihedral. 1You should be comfortable with both multiplicative and additive notation. 2More generally any substructure. 1 Cosets and Factor Groups Definition. -
1 Sets and Classes
Prerequisites Topological spaces. A set E in a space X is σ-compact if there exists a S1 sequence of compact sets such that E = n=1 Cn. A space X is locally compact if every point of X has a neighborhood whose closure is compact. A subset E of a locally compact space is bounded if there exists a compact set C such that E ⊂ C. Topological groups. The set xE [or Ex] is called a left translation [or right translation.] If Y is a subgroup of X, the sets xY and Y x are called (left and right) cosets of Y . A topological group is a group X which is a Hausdorff space such that the transformation (from X ×X onto X) which sends (x; y) into x−1y is continuous. A class N of open sets containing e in a topological group is a base at e if (a) for every x different form e there exists a set U in N such that x2 = U, (b) for any two sets U and V in N there exists a set W in N such that W ⊂ U \ V , (c) for any set U 2 N there exists a set W 2 N such that V −1V ⊂ U, (d) for any set U 2 N and any element x 2 X, there exists a set V 2 N such that V ⊂ xUx−1, and (e) for any set U 2 N there exists a set V 2 N such that V x ⊂ U. If N is a satisfies the conditions described above, and if the class of all translation of sets of N is taken for a base, then, with respect to the topology so defined, X becomes a topological group. -
On Families of Mutually Exclusive Sets
ANNALS OF MATHEMATICS Vol. 44, No . 2, April, 1943 ON FAMILIES OF MUTUALLY EXCLUSIVE SETS BY P . ERDÖS AND A. TARSKI (Received August 11, 1942) In this paper we shall be concerned with a certain particular problem from the general theory of sets, namely with the problem of the existence of families of mutually exclusive sets with a maximal power . It will turn out-in a rather unexpected way that the solution of these problems essentially involves the notion of the so-called "inaccessible numbers ." In this connection we shall make some general remarks regarding inaccessible numbers in the last section of our paper . §1. FORMULATION OF THE PROBLEM . TERMINOLOGY' The problem in which we are interested can be stated as follows : Is it true that every field F of sets contains a family of mutually exclusive sets with a maximal power, i .e . a family O whose cardinal number is not smaller than the cardinal number of any other family of mutually exclusive sets contained in F . By a field of sets we understand here as usual a family F of sets which to- gether with every two sets X and Y contains also their union X U Y and their difference X - Y (i.e. the set of those elements of X which do not belong to Y) among its elements . A family O is called a family of mutually exclusive sets if no set X of X of O is empty and if any two different sets of O have an empty inter- section. A similar problem can be formulated for other families e .g . -
Problems from Ring Theory
Home Page Problems From Ring Theory In the problems below, Z, Q, R, and C denote respectively the rings of integers, JJ II rational numbers, real numbers, and complex numbers. R generally denotes a ring, and I and J usually denote ideals. R[x],R[x, y],... denote rings of polynomials. rad R is defined to be {r ∈ R : rn = 0 for some positive integer n} , where R is a ring; rad R is called the nil radical or just the radical of R . J I Problem 0. Let b be a nilpotent element of the ring R . Prove that 1 + b is an invertible element Page 1 of 14 of R . Problem 1. Let R be a ring with more than one element such that aR = R for every nonzero Go Back element a ∈ R. Prove that R is a division ring. Problem 2. If (m, n) = 1 , show that the ring Z/(mn) contains at least two idempotents other than Full Screen the zero and the unit. Problem 3. If a and b are elements of a commutative ring with identity such that a is invertible Print and b is nilpotent, then a + b is invertible. Problem 4. Let R be a ring which has no nonzero nilpotent elements. Prove that every idempotent Close element of R commutes with every element of R. Quit Home Page Problem 5. Let A be a division ring, B be a proper subring of A such that a−1Ba ⊆ B for all a 6= 0 . Prove that B is contained in the center of A . -
On Semilattice Structure of Mizar Types
FORMALIZED MATHEMATICS Volume 11, Number 4, 2003 University of Białystok On Semilattice Structure of Mizar Types Grzegorz Bancerek Białystok Technical University Summary. The aim of this paper is to develop a formal theory of Mizar types. The presented theory is an approach to the structure of Mizar types as a sup-semilattice with widening (subtyping) relation as the order. It is an abstrac- tion from the existing implementation of the Mizar verifier and formalization of the ideas from [9]. MML Identifier: ABCMIZ 0. The articles [20], [14], [24], [26], [23], [25], [3], [21], [1], [11], [12], [16], [10], [13], [18], [15], [4], [2], [19], [22], [5], [6], [7], [8], and [17] provide the terminology and notation for this paper. 1. Semilattice of Widening Let us mention that every non empty relational structure which is trivial and reflexive is also complete. Let T be a relational structure. A type of T is an element of T . Let T be a relational structure. We say that T is Noetherian if and only if: (Def. 1) The internal relation of T is reversely well founded. Let us observe that every non empty relational structure which is trivial is also Noetherian. Let T be a non empty relational structure. Let us observe that T is Noethe- rian if and only if the condition (Def. 2) is satisfied. (Def. 2) Let A be a non empty subset of T . Then there exists an element a of T such that a ∈ A and for every element b of T such that b ∈ A holds a 6< b. -
Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R. -
Completely Representable Lattices
Completely representable lattices Robert Egrot and Robin Hirsch Abstract It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be jCRL ∩ mCRL. We prove CRL ⊂ biCRL = mCRL ∩ jCRL ⊂ mCRL =6 jCRL ⊂ DL where the marked inclusions are proper. Let L be a DL. Then L ∈ mCRL iff L has a distinguishing set of complete, prime filters. Similarly, L ∈ jCRL iff L has a distinguishing set of completely prime filters, and L ∈ CRL iff L has a distinguishing set of complete, completely prime filters. Each of the classes above is shown to be pseudo-elementary hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary. 1 Introduction An atomic representation h of a Boolean algebra B is a representation h: B → ℘(X) (some set X) where h(1) = {h(a): a is an atom of B}. It is known that a representation of a Boolean algebraS is a complete representation (in the sense of a complete embedding into a field of sets) if and only if it is an atomic repre- sentation and hence that the class of completely representable Boolean algebras is precisely the class of atomic Boolean algebras, and hence is elementary [6]. arXiv:1201.2331v3 [math.RA] 30 Aug 2016 This result is not obvious as the usual definition of a complete representation is thoroughly second order. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Optimisation
Optimisation Embryonic notes for the course A4B33OPT This text is incomplete and may be added to and improved during the semester. This version: 25th February 2015 Tom´aˇsWerner Translated from Czech to English by Libor Spaˇcekˇ Czech Technical University Faculty of Electrical Engineering Contents 1 Formalising Optimisation Tasks8 1.1 Mathematical notation . .8 1.1.1 Sets ......................................8 1.1.2 Mappings ...................................9 1.1.3 Functions and mappings of several real variables . .9 1.2 Minimum of a function over a set . 10 1.3 The general problem of continuous optimisation . 11 1.4 Exercises . 13 2 Matrix Algebra 14 2.1 Matrix operations . 14 2.2 Transposition and symmetry . 15 2.3 Rankandinversion .................................. 15 2.4 Determinants ..................................... 16 2.5 Matrix of a single column or a single row . 17 2.6 Matrixsins ...................................... 17 2.6.1 An expression is nonsensical because of the matrices dimensions . 17 2.6.2 The use of non-existent matrix identities . 18 2.6.3 Non equivalent manipulations of equations and inequalities . 18 2.6.4 Further ideas for working with matrices . 19 2.7 Exercises . 19 3 Linearity 22 3.1 Linear subspaces . 22 3.2 Linearmapping.................................... 22 3.2.1 The range and the null space . 23 3.3 Affine subspace and mapping . 24 3.4 Exercises . 25 4 Orthogonality 27 4.1 Scalarproduct..................................... 27 4.2 Orthogonal vectors . 27 4.3 Orthogonal subspaces . 28 4.4 The four fundamental subspaces of a matrix . 28 4.5 Matrix with orthonormal columns . 29 4.6 QR decomposition . 30 4.6.1 (?) Gramm-Schmidt orthonormalisation . -
Quasigroups, Right Quasigroups, Coverings and Representations Tengku Simonthy Renaldin Fuad Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1993 Quasigroups, right quasigroups, coverings and representations Tengku Simonthy Renaldin Fuad Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Fuad, Tengku Simonthy Renaldin, "Quasigroups, right quasigroups, coverings and representations " (1993). Retrospective Theses and Dissertations. 10427. https://lib.dr.iastate.edu/rtd/10427 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. -
The Quasigroup Block Cipher and Its Analysis Matthew .J Battey University of Nebraska at Omaha
University of Nebraska at Omaha DigitalCommons@UNO Student Work 5-2014 The Quasigroup Block Cipher and its Analysis Matthew .J Battey University of Nebraska at Omaha Follow this and additional works at: https://digitalcommons.unomaha.edu/studentwork Part of the Computer Sciences Commons Recommended Citation Battey, Matthew J., "The Quasigroup Block Cipher and its Analysis" (2014). Student Work. 2892. https://digitalcommons.unomaha.edu/studentwork/2892 This Thesis is brought to you for free and open access by DigitalCommons@UNO. It has been accepted for inclusion in Student Work by an authorized administrator of DigitalCommons@UNO. For more information, please contact [email protected]. The Quasigroup Block Cipher and its Analysis A Thesis Presented to the Department of Computer Sience and the Faculty of the Graduate College University of Nebraska In partial satisfaction of the requirements for the degree of Masters of Science by Matthew J. Battey May, 2014 Supervisory Committee: Abhishek Parakh, Co-Chair Haifeng Guo, Co-Chair Kenneth Dick Qiuming Zhu UMI Number: 1554776 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 1554776 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. -
Section 17.5. the Carathéodry-Hahn Theorem: the Extension of A
17.5. The Carath´eodory-Hahn Theorem 1 Section 17.5. The Carath´eodry-Hahn Theorem: The Extension of a Premeasure to a Measure Note. We have µ defined on S and use µ to define µ∗ on X. We then restrict µ∗ to a subset of X on which µ∗ is a measure (denoted µ). Unlike with Lebesgue measure where the elements of S are measurable sets themselves (and S is the set of open intervals), we may not have µ defined on S. In this section, we look for conditions on µ : S → [0, ∞] which imply the measurability of the elements of S. Under these conditions, µ is then an extension of µ from S to M (the σ-algebra of measurable sets). ∞ Definition. A set function µ : S → [0, ∞] is finitely additive if whenever {Ek}k=1 ∞ is a finite disjoint collection of sets in S and ∪k=1Ek ∈ S, we have n n µ · Ek = µ(Ek). k=1 ! k=1 [ X Note. We have previously defined finitely monotone for set functions and Propo- sition 17.6 gives finite additivity for an outer measure on M, but this is the first time we have discussed a finitely additive set function. Proposition 17.11. Let S be a collection of subsets of X and µ : S → [0, ∞] a set function. In order that the Carath´eodory measure µ induced by µ be an extension of µ (that is, µ = µ on S) it is necessary that µ be both finitely additive and countably monotone and, if ∅ ∈ S, then µ(∅) = 0.