ON SOME CLASSES OF GOOD QUOTIENT RELATIONS

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Novi Sad J. Math. Vol. 32, No. 2, 2002, ON SOME CLASSES OF GOOD QUOTIENT RELATIONS Ivica Bošnjak 1, Rozália Madarász 1 Abstract. The notion of a good quotient relation has been introduced as an attempt to generalize the notion of a quotient algebra to relations on an algebra which are not necessarily congruence relations. In this paper we investigate some special classes of good relations for which the generalized versions of the well-known isomorphism theorems can be proved. AMS Mathematics Subject Classification (2000): 08A30,08A99 Key words and phrases: power algebra, good relation, quasi-congruence, isomorphism theorems 1. Introduction The notion of a generalized quotient algebra and the corresponding notion of a good (quotient) relation has been introduced in [6] and [7] as an attempt to generalize the notion of a quotient algebra to relations on an algebra which are not necessarily congruences. From Definition 1 it is easy to see that every non-trivial algebra has good relations which are not congruences. Up to now, most of the results on good relations have been obtained in the context of power structures (see [4] for an overview on power structures). One of the reasons is that for any relation R A 2, the generalized quotient set A/R is a subset of the power set P(A). A power version of the well-known homomorphism theorem was proved in [4]. This theorem was the motivation for introducing the notions of very good ([6],[7]), Hoare good and Smyth good relations ([5]). For these special good relations some versions of the power homomorphism theorems were proved in [5]. In [2] the relationships between Hoare good, Smyth good and very good relations were described. In [3], these sets of special good relations have been investigated with respect to the settheoretical operations and various ways of powering. In the present paper we introduce and study some special good relations for which the well-known isomorphism theorems can be proved. These sets of good relations are also studied from the lattice-theoretical point of view. 1 Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad, Yugoslavia

2 132 I. Bošnjak, R. Madarász 2. Definitions and the isomorphism theorems Definition 1. ([6], [7]) Let A = A, F be an algebra and R A 2. (1) For any a A we define a/r = {b bra}. The corresponding generalized quotient set is A/R = {a/r a A}. (2) Relation ε(r) A 2 is defined by: (a, b) ε(r) a/r = b/r. (3) We call R a good (quotient) relation on A if ε(r) is a congruence on A. The set of all good relations on A we denote by G(A). (4) If R is a good quotient relation on A, the corresponding generalized quotient algebra A/R is A/R = A/R, { f f F }, where operations f (f F ) are defined in the following way: if ar(f) = n, then for any a 1,..., a n A f (a 1 /R,..., a n /R) = f(a 1,..., a n )/R. Some examples of good relations are: partial orders, compatible quasiorders, structure preserving relations (see [4]), and quasi-congruences (see Definition 2). In [3] the set G(A) of all good relations on an algebra A is described and a necessary and sufficient condition is given for the partially ordered set G(A) = G(A), to be a lattice. It can be easily proved that the following extended homomorphism theorem holds for good quotient relations. Theorem 1. Let A and B be algebras of the same type. Then B is a homomorphic image of A if and only if there is a good relation R on A such that B = A/R. In [4] and [5] some power versions of the homomorphism theorem were proved (for congruences and for Hoare good, Smyth good and very good relations). In [1] some isomorphism theorems for the so-called regular power rings can be found. Unfortunately, the well-known isomorphism theorems cannot be generalized to the whole class of good relations. Example 1. Let A = {0, 1, 2, 3, 4}, R = {(0, 2), (1, 3), (2, 4), (3, 1), (4, 0)}, f(0) = 1, f(1) = 2, f(2) = 0, f(3) = 4, f(4) = 3. Then R is a good relation on A = A, f, while for B = {0, 1, 2}, R B = R B 2 is not a good relation on subalgebra B of A.

3 On some classes of good quotient relations 133 Example 2. Let A = A, F be an algebra with all relations on it being good (such algebras are described in [3]), and a, b, c A. If R = {(a, a), (a, b), (a, c)}, then (with the standard definition of a quotient relation) relation R/R is not well defined since (a, c) R (a/r, c/r) R/R and, on the other hand (b, c) / R (a/r, c/r) = (b/r, c/r) / R/R. In the sequel, we are searching for smaller classes of good relations for which some generalized versions of the isomorphism theorems could be proved. The following classes proved to be suitable. Definition 2. Let A be an algebra and R A 2. (1) ([5]) We call R a quasi-equivalence on A if for all x,y A x/r = y/r xry & yrx. We denote the set of all quasi-equivalences on A by QEqvA. (2) We call R a quasi-congruence on A if R is a good quasi-equivalence. The set of all quasi-congruences on A we denote by QConA. Let us note that the expression quasi-congruence in [5] is used for compatible quasi-equivalences (which are also good relations). We will call these quasi-congruences B-quasi-congruences (Brink s quasi-congruences). The set of all B-quasi-congruences on A we will denote by BQConA. Definition 3. Let A be an algebra and R QConA. We call R a two-side quasi-congruence on A if for all x,y,z A xry & yrx & xrz yrz. The set of all two-side quasi-congruences on A we denote by T SQConA. Example 3. Every reflexive and transitive relation on A is a quasi-equivalence on A. Every reflexive and anti-symmetric relation on A is a two-side quasicongruence on arbitrary algebra A with the carrier set A. In general, T SQConA is a proper subset of QConA and incomparable with BQConA. Example 4. Let A = {a, b, c}, f(a) = f(b) = a, f(c) = c, A = A, f, R = {(a, a), (b, a), (a, b), (b, b), (a, c), (c, c)}, S = {(a, a), (b, b), (b, c), (c, c)}. Then R BQConA \ T SQConA and S T SQConA \ BQConA. Lemma 1. Let A = A, F be an algebra and R QEqvA. conditions are equivalent: The following (1) R QConA

4 134 I. Bošnjak, R. Madarász (2) For any f F, if ar(f) = n then for any x 1,..., x n, y 1,..., y n A ( i n)(x i Ry i & y i Rx i ) (f(x 1,..., x n )Rf(y 1,..., y n ) & f(y 1,..., y n )Rf(x 1,..., x n )) Definition 4. Let A be an algebra, B a subuniverse of A, R A 2. We define B R A as B R = {a A ( b B) a/r = b/r}. Lemma 2. Let A be an algebra and B a subalgebra of A. Then a) If R QConA then R B QConB. b) If R G(A) then B R is a subuniverse of A. If R G(A) and B A then the subalgebra of A with the universe B R we denote by B R. Theorem 2. Let A be an algebra, B A and R QConA. Then B/R B = B R /R B R. Proof. According to Lemma 2, algebras B/R B and B R /R B R are well defined. Let Φ : B/R B B R /R B R be a mapping defined in the natural way: Φ(b/R B ) = b/r B R. It is easy to prove that Φ is an isomorphism. Definition 5. Let A be an algebra, R, S T SQConA, R S. The relation S/R A/R A/R is defined as (a/r, b/r) S/R (a, b) S. Lemma 3. Let A be an algebra, R, S T SQConA, R S. T SQCon(A/R). Then S/R Theorem 3. Let A be an algebra, R, S T SQConA, R S. Then / A/R S/R = A/S. / Proof. According to Lemma 3, the algebra A/R S/R is well defined. Let us / define a mapping Φ : A/R S/R A/S in the natural way: / Φ(a/R S/R) = a/s. Then it is easy to prove that Φ is an isomorphism.

5 On some classes of good quotient relations The lattices of quasi-congruences In [5] it is proved that for any algebra A, the set BQConA is closed under arbitrary intersections. Hence BQConA, is a complete lattice. It is easy to see that the same is true for the sets QEqvA, QConA and T SQConA. We can prove even more. Theorem 4. Let A be an algebra with the carrier set A. Then QEqvA,, QConA,, BQConA,, T SQConA, are algebraic lattices. Proof. Let A be an algebra of type F. In all cases we will define an algebra B with the universe A A such that SubB (the set of all subuniverses of B) will be equal to the corresponding set of quasi-equivalences. Since SubB, is an algebraic lattice, the proposition holds. Case QEqvA, : It is not hard to see that R A 2 is a quasi-equivalence if and only if R is reflexive and for all x,y,z A it holds xry & yrx & zrx zry. Hence the fundamental operations of B will be: - the nullary operations (a, a) for every a A; - a ternary operation t : B 3 B defined by { (c1, a t((a 1, a 2 ), (b 1, b 2 ), (c 1, c 2 )) = 2 ) if a 1 = b 2 = c 2 & a 2 = b 1 (a 1, a 2 ) else Then SubB = QEqvA and QEqvA, is an algebraic lattice. Case BQConA, : The fundamental operations of the algebra B are all operations defined in the previous case and for all f F operations f B defined in the following way: if ar(f) = n, then ar(f B ) = n and for all a 1,..., a n, b 1,..., b n A it holds f B ((a 1, b 1 ),..., (a n, b n )) = (f A (a 1,..., a n ), f A (b 1,..., b n )). Then SubB = BQConA and BQConA, is an algebraic lattice. Case QConA, : The fundamental operations of the algebra B are all operations defined in the case QEqvA and for all f F operations f1 B, f2 B defined in the following way: if ar(f) = n, then ar(f1 B ) = ar(f2 B ) = 2n and for all a 1,..., a n, b 1,..., b n, c 1,..., c n, d 1,..., d n A it holds f B 1 ((a 1, b 1 ),..., (a n, b n ), (c 1, d 1 ),..., (c n, d n )) =

6 136 I. Bošnjak, R. Madarász { (f A (a 1,..., a n ), f A (b 1,..., b n )) if c i = b i & d i = a i for all i n (a 1, b 1 ) else f B 2 ((a 1, b 1 ),..., (a n, b n ), (c 1, d 1 ),..., (c n, d n )) = { (f A (b 1,..., b n ), f A (a 1,..., a n )) if c i = b i & d i = a i for all i n (a 1, b 1 ) else According to Lemma 1, SubB = QConA and QConA, is an algebraic lattice. Case T SQConA, : The fundamental operations of the algebra B are all operations defined in the previous case and a ternary operation t defined in the following way: { t (a2, c ((a 1, a 2 ), (b 1, b 2 ), (c 1, c 2 )) = 2 ) if b 1 = a 2 & b 2 = c 1 = a 1 (a 1, a 2 ) else Then SubB = T SQConA and T SQConA, is an algebraic lattice. Let us note that G(A), is almost never a lattice (see [3]). In [5] it is proved that ConA, is a sublattice of BQConA. We can prove that it is also a sublattice of QEqvA,QConA and T SQConA. Lemma 4. ([5]) (1) For every R,S QEqvA it holds R S ε(r) ε(s). (2) For every R QEqvA it holds ε(r) R. Theorem 5. Let Σ be a set of quasi-congruences of some algebra A such that ConA Σ and Σ is closed under (finite) intersections. If Σ, is a lattice, then ConA, is a sublattice of Σ,. Proof. Since Σ is closed under intersections, the infimum in Σ, coincides with the intersection, like in ConA. Therefore, we only have to prove that supremums coincide in these two lattices. Let ρ, σ ConA and θ = sup Σ (ρ, σ). Let us note that it is sufficient to prove that θ is a congruence relation, for this implies θ = sup ConA (ρ, σ). According to Lemma 4(1), since ρ θ and σ θ and all these relations are quasi-equivalences, it holds ε(ρ) ε(θ) and ε(σ) ε(θ). But ρ,σ ConA which implies ε(ρ) = ρ and ε(σ) = σ, i.e. ρ ε(θ) and σ ε(θ). Since θ QConA G(A), we have ε(θ) ConA, which implies θ ε(θ). According to Lemma 4(2) it holds ε(θ) θ, so ε(θ) = θ which means θ ConA. Corollary 1. For any algebra A, ConA, is a sublattice of the following lattices: QConA,, BQConA,, T SQConA,.

7 On some classes of good quotient relations 137 Proof. Follows from Theorem 2. Corollary 2. For any algebra A = A, F, ConA, is a sublattice of QEqvA,. Proof. Let B be an algebra with the carrier set A such that all fundamental operations of B are projections. Then ConB = EqvA and BQConB = QEqvA. Therefore, according to Corollary 1, EqvA is a sublattice of QEqvA. Since ConA is a sublattice of EqvA for any algebra A = A, F, this implies that ConA is a sublattice of QEqvA. The above two corollaries cover all the cases when one of the lattices of quasi-congruences is a sublattice of another. Example 5. Let A = {a, b, c}, f(a) = f(c) = c,f(b) = b, A = A, f, R = {(a, b)}, S = {(b, a)}. Then R,S,R S QEqvA and sup(r, S) = R S. On the other hand, R,S QConA but R S / QConA, for (a, b) R S, (b, a) R S but (f(a), f(b)) / R S. Example 6. Let A = {a, b, c, d, 1, 2, 3, 4}, A = A, f where f : A 2 A is defined in the following way: f(a, c) = 1, f(b, c) = 2, f(a, d) = 3, f(b, d) = 4 and f(x, y) = 1 for all other (x, y) A 2. If R = {(a, b), (1, 2), (3, 4)}, S = {(c, d), (1, 3), (2, 4)}, then R, S BQConA, R S QEqvA, but R S / BQConA because (a, b) R S, (c, d) R S, but (f(a, c), f(b, d)) = (1, 4) / R S. Theorem 6. (1) Lattices QConA,BQConA,T SQConA are not necessarily sublattices of QEqvA. (2) Lattices BQConA and T SQConA are not necessarily sublattices of QConA. Proof. (1) Example 5 shows that QConA does not have to be a sublattice of QEqvA. This example also shows that T SQConA is not necessarily sublattice of QEqvA for R,S,R S QEqvA, R,S T SQConA, but R S / T SQConA. Example 6 shows that BQConA does not have to be a sublattice of QEqvA. (2) Example 6 also shows that BQConA is not necessarily a sublattice of QConA for R,S BQConA, R S QConA, but R S / BQConA. And finally, let A be an algebra from Example 6 and R = {(a, b), (1, 2), (3, 4)}, S = {c, d), (1, 3), (2, 4), (2, 1)}. Then R,S T SQConA, R S QConA, but R S / T SQConA because (1, 2) R S, (2, 1) R S, (1, 3) R S and (2, 3) / R S.

8 138 I. Bošnjak, R. Madarász There exists algebras A = A, F such that QEqvA = QConA, or QEqvA = BQConA and it is not difficult to describe them. Theorem 7. Let A be an algebra and A 3. The following conditions are equivalent: (1) QEqvA = QConA, (2) QEqvA = BQConA, (3) EqvA = ConA. Proof. (1) (3) If QEqvA = QConA then QEqvA G(A) and consequently EqvA G(A). This implies ε(r) is a congruence for every R EqvA, and since ε(r) = R, we conclude EqvA = ConA. (3) (2) Let EqvA = ConA. It is known that all the fundamental operations of A have to be projections or constant operations. Then, every reflexive relation is compatible on A, so QEqvA = BQConA. (2) (1) Obvious. If A = 2, conditions (1) and (3) hold for any algebra A, but (2) is not always true. Another problem that might be interesting is to describe algebras A such that BQConA = ConA (B-quasi-congruence-trivial algebras). Of course, one can easily notice that for any non-trivial algebra A, QConA ConA and T SQConA = ConA. Lemma 5. Let A be an algebra which has a term p(x, y, z) such that at least two of the following identities are satisfied on A: Then BQConA = ConA. p(x, x, y) y, p(x, y, x) y, p(y, x, x) y. Proof. It is easy to see that every symmetric quasi-equivalence is an equivalence relation as well. Now, suppose, for example that p(x, x, y) p(x, y, x) y holds on A. Then, for every R BQConA we have (xry & xrx & yry) p(x, x, y)rp(y, x, y) yrx. Hence R ConA. The proof is analogous in two remaining cases. Theorem 8. Let A be an algebra which has a Mal cev term (or a minority term). Then BQConA = ConA.

9 On some classes of good quotient relations 139 Proof. If p(x, y, z) is a Mal cev term on A then A satisfies identities p(x, x, y) y, p(x, y, y) x. Therefore, the conditions of Lemma 3 are satisfied. A term m(x, y, z) is a minority term on A if A satisfies identities m(x, x, y) m(x, y, x) m(y, x, x) y and the theorem is a consequence of Lemma 3 again. Corollary 3. Let V be a congruence permutable variety. Then for any A V, BQConA = ConA. Proof. Follows from Theorem 6 and the well-known theorem of Mal cev on the congruence permutable varieties. The following example shows that there exist B-quasi-congruence-trivial algebras that are not congruence permutable. Example 7. Let A = {a, b, c}, f(a) = f(c) = b, f(b) = a, g(a) = g(b) = c, g(c) = a, A = A, {f, g}. If R BQConA, the following is obvious: Also, if (b, c) R then (1) (2) (3) (a, b) R (b, a) R, (a, c) R (c, a) R. (f(b), f(c)) = (a, b) R; (g(b), g(c)) = (c, a) R; (f 2 (b), f 2 (c)) = (b, a) R. It follows directly from (1),(2),(3) and the definition of quasi-equivalence that (c, b) R. We can prove in a similar way that (c, b) R (b, c) R. Thus every B-quasi-congruence on A is symmetric, which implies A is B-quasicongruence-trivial. Since ConA is a 4-element set, it is easy to check that A is not congruence permutable. Example 8. Let A = {a, b},, be a two-element chain. Then R = {(a, a), (a, b), (b, b)} is a B-quasi-congruence which is not a congruence relation. Hence the variety of all lattices is an example of a congruence-distributive variety which is not B-quasi-congruence trivial. The following question remains to be answered: is it true that for any algebraic lattice L, there is an algebra A such that L = BQConA? References [1] Bingxue, Y., Isomorphic theorems of regular power rings, Italian Journal of Pure and Applied Mathematics, to appear. [2] Bošnjak, I., R.Madarász, Power algebras and generalized quotient algebras, Algebra univers. 45 (2001)

10 140 I. Bošnjak, R. Madarász [3] Bošnjak I., R.Madarász, Good quotient relations and power algebras, Novi Sad J. Math. Vol. 29, No. 2 (1999), 71-84, Proc. VIII Int. Conf. Algebra & Logic (Novi Sad, 1998). [4] Brink, C., Power structures, Alg. Universalis, 30 (1993), [5] Brink, C., D.F.Jacobs, K.Nelte, R.Sekaran, Generalized quotient algebras and power algebras, 1996, 1-11, manuscript. [6] Goranko, V., Notes on Power Structures Section 7 (Generalized Quotient Algebras), 1-3, 1994, private communication. [7] Madarász, R., Generalized factor algebras, Matem. Bilten, 18(XLIV), 1994, Received by the editors September 3, 1999.

Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

Discrete Mathematics. Hans Cuypers. October 11, 2007

Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

Some Special Artex Spaces Over Bi-monoids

Some Special Artex Spaces Over Bi-monoids K.Muthukumaran (corresponding auther) Assistant Professor PG and Research Department Of Mathematics, Saraswathi Narayanan College, Perungudi Madurai-625022,Tamil

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

Rough Semi Prime Ideals and Rough Bi-Ideals in Rings

Int Jr. of Mathematical Sciences & Applications Vol. 4, No.1, January-June 2014 Copyright Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com Rough Semi Prime Ideals and Rough Bi-Ideals in

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

Solutions to In-Class Problems Week 4, Mon.

Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics for Computer Science September 26 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 26, 2005, 1050 minutes Solutions

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS

ON THE EMBEDDING OF BIRKHOFF-WITT RINGS IN QUOTIENT FIELDS DOV TAMARI 1. Introduction and general idea. In this note a natural problem arising in connection with so-called Birkhoff-Witt algebras of Lie

The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

Finite Projective demorgan Algebras. honoring Jorge Martínez

Finite Projective demorgan Algebras Simone Bova Vanderbilt University (Nashville TN, USA) joint work with Leonardo Cabrer March 11-13, 2011 Vanderbilt University (Nashville TN, USA) honoring Jorge Martínez

Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

Chapter 7. Homotopy. 7.1 Basic concepts of homotopy. Example: z dz. z dz = but

Chapter 7 Homotopy 7. Basic concepts of homotopy Example: but γ z dz = γ z dz γ 2 z dz γ 3 z dz. Why? The domain of /z is C 0}. We can deform γ continuously into γ 2 without leaving C 0}. Intuitively,

Non-unique factorization of polynomials over residue class rings of the integers

Comm. Algebra 39(4) 2011, pp 1482 1490 Non-unique factorization of polynomials over residue class rings of the integers Christopher Frei and Sophie Frisch Abstract. We investigate non-unique factorization

INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

Invertible elements in associates and semigroups. 1

Quasigroups and Related Systems 5 (1998), 53 68 Invertible elements in associates and semigroups. 1 Fedir Sokhatsky Abstract Some invertibility criteria of an element in associates, in particular in n-ary

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

Elements of Abstract Group Theory

Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

University of Oslo MAT2 Project The Banach-Tarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the Banach-Tarski paradox states that for any ball in R, it is possible

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)

ON SOME CLASSES OF REGULAR ORDER SEMIGROUPS

Commun. Korean Math. Soc. 23 (2008), No. 1, pp. 29 40 ON SOME CLASSES OF REGULAR ORDER SEMIGROUPS Zhenlin Gao and Guijie Zhang Reprinted from the Communications of the Korean Mathematical Society Vol.

Four Unsolved Problems in Congruence Permutable Varieties

Four Unsolved Problems in Congruence Permutable Varieties Ross Willard University of Waterloo, Canada Nashville, June 2007 Ross Willard (Waterloo) Four Unsolved Problems Nashville, June 2007 1 / 27 Congruence

Irreducible Representations of Wreath Products of Association Schemes

Journal of Algebraic Combinatorics, 18, 47 52, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Irreducible Representations of Wreath Products of Association Schemes AKIHIDE HANAKI

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

The Division Algorithm for Polynomials Handout Monday March 5, 2012

The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,

SMALL SKEW FIELDS CÉDRIC MILLIET

SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite

Interpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences

DISCUSSION PAPER SERIES IZA DP No. 2594 Interpersonal Comparisons of Utility: An Algebraic Characterization of Projective Preorders and Some Welfare Consequences Juan Carlos Candeal Esteban Induráin José

On the Algebraic Structures of Soft Sets in Logic

Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 1873-1881 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department

RINGS WITH A POLYNOMIAL IDENTITY

RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in

Bi-approximation Semantics for Substructural Logic at Work

Bi-approximation Semantics for Substructural Logic at Work Tomoyuki Suzuki Department of Computer Science, University of Leicester Leicester, LE1 7RH, United Kingdom tomoyuki.suzuki@mcs.le.ac.uk Abstract

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

ON ROUGH (m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS

U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 1, 2013 ISSN 1223-7027 ON ROUGH m, n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS Naveed Yaqoob 1, Muhammad Aslam 1, Bijan Davvaz 2, Arsham Borumand Saeid 3 In this

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

On an algorithm for classification of binary self-dual codes with minimum distance four

Thirteenth International Workshop on Algebraic and Combinatorial Coding Theory June 15-21, 2012, Pomorie, Bulgaria pp. 105 110 On an algorithm for classification of binary self-dual codes with minimum

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS

Bulletin of the Section of Logic Volume 15/2 (1986), pp. 72 79 reedition 2005 [original edition, pp. 72 84] Ryszard Ladniak ON THE DEGREE OF MAXIMALITY OF DEFINITIONALLY COMPLETE LOGICS The following definition

CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

Two classes of ternary codes and their weight distributions

Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight

CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

FIXED POINT SETS OF FIBER-PRESERVING MAPS

FIXED POINT SETS OF FIBER-PRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 e-mail: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics

PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

Generating Direct Powers

Generating Direct Powers Nik Ruškuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, Novi Sad, 16 March 2012 Algebraic structures Classical: groups, rings, modules, algebras, Lie algebras. Semigroups.

Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

s-convexity, model sets and their relation

s-convexity, model sets and their relation Zuzana Masáková Jiří Patera Edita Pelantová CRM-2639 November 1999 Department of Mathematics, Faculty of Nuclear Science and Physical Engineering, Czech Technical

7. Some irreducible polynomials

7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

4.1 Modules, Homomorphisms, and Exact Sequences

Chapter 4 Modules We always assume that R is a ring with unity 1 R. 4.1 Modules, Homomorphisms, and Exact Sequences A fundamental example of groups is the symmetric group S Ω on a set Ω. By Cayley s Theorem,

Introduction to Finite Fields (cont.)

Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number

Transfer of the Ramsey Property between Classes

1 / 20 Transfer of the Ramsey Property between Classes Lynn Scow Vassar College BLAST 2015 @ UNT 2 / 20 Classes We consider classes of finite structures such as K < = {(V,

3. SETS, FUNCTIONS & RELATIONS

3. SETS, FUNCTIONS & RELATIONS If I see the moon, then the moon sees me 'Cos seeing's symmetric as you can see. If I tell Aunt Maude and Maude tells the nation Then I've told the nation 'cos the gossiping

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES THEODORE A. SLAMAN AND ANDREA SORBI Abstract. We show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: In fact, below

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

Biinterpretability up to double jump in the degrees

Biinterpretability up to double jump in the degrees below 0 0 Richard A. Shore Department of Mathematics Cornell University Ithaca NY 14853 July 29, 2013 Abstract We prove that, for every z 0 0 with z

Cyclotomic Extensions

Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate

1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT

Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally

Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

ON UNIQUE FACTORIZATION DOMAINS

ON UNIQUE FACTORIZATION DOMAINS JIM COYKENDALL AND WILLIAM W. SMITH Abstract. In this paper we attempt to generalize the notion of unique factorization domain in the spirit of half-factorial domain. It

Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

Generic Polynomials of Degree Three

Generic Polynomials of Degree Three Benjamin C. Wallace April 2012 1 Introduction In the nineteenth century, the mathematician Évariste Galois discovered an elegant solution to the fundamental problem

9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

SECRET sharing schemes were introduced by Blakley [5]

206 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 1, JANUARY 2006 Secret Sharing Schemes From Three Classes of Linear Codes Jin Yuan Cunsheng Ding, Senior Member, IEEE Abstract Secret sharing has

Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.

Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES PAUL SHAFER Abstract. We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order

About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range

THEORY OF COMPUTING, Volume 1 (2005), pp. 37 46 http://theoryofcomputing.org Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range Andris Ambainis

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

x < y iff x < y, or x and y are incomparable and x χ(x,y) < y χ(x,y).

12. Large cardinals The study, or use, of large cardinals is one of the most active areas of research in set theory currently. There are many provably different kinds of large cardinals whose descriptions

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

Putnam Notes Polynomials and palindromes

Putnam Notes Polynomials and palindromes Polynomials show up one way or another in just about every area of math. You will hardly ever see any math competition without at least one problem explicitly concerning

Turing Degrees and Definability of the Jump. Theodore A. Slaman. University of California, Berkeley. CJuly, 2005

Turing Degrees and Definability of the Jump Theodore A. Slaman University of California, Berkeley CJuly, 2005 Outline Lecture 1 Forcing in arithmetic Coding and decoding theorems Automorphisms of countable

Finitely Additive Dynamic Programming and Stochastic Games. Bill Sudderth University of Minnesota

Finitely Additive Dynamic Programming and Stochastic Games Bill Sudderth University of Minnesota 1 Discounted Dynamic Programming Five ingredients: S, A, r, q, β. S - state space A - set of actions q(

Incorporating Evidence in Bayesian networks with the Select Operator

Incorporating Evidence in Bayesian networks with the Select Operator C.J. Butz and F. Fang Department of Computer Science, University of Regina Regina, Saskatchewan, Canada SAS 0A2 {butz, fang11fa}@cs.uregina.ca

arxiv:1312.1919v3 [math.gr] 22 Feb 2015

MOUFANG SEMIDIRECT PRODUCTS OF LOOPS WITH GROUPS AND INVERSE PROPERTY EXTENSIONS MARK GREER AND LEE RANEY arxiv:1312.1919v3 [math.gr] 22 Feb 2015 Abstract. We investigate loops which can be written as

G = G 0 > G 1 > > G k = {e}

Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

P NP for the Reals with various Analytic Functions

P NP for the Reals with various Analytic Functions Mihai Prunescu Abstract We show that non-deterministic machines in the sense of [BSS] defined over wide classes of real analytic structures are more powerful

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction

ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that