Tropical matrix algebra
|
|
- Myra Rodgers
- 6 years ago
- Views:
Transcription
1 Tropical matrix algebra Marianne Johnson 1 & Mark Kambites 2 NBSAN, University of York, 25th November University of Manchester. Supported by CICADA (EPSRC grant EP/E050441/1). 2 University of Manchester. Supported by an RCUK Academic Fellowship. Johnson & Kambites Tropical matrix algebra 1 / 17
2 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Johnson & Kambites Tropical matrix algebra 2 / 17
3 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Properties R is an idempotent semifield: Johnson & Kambites Tropical matrix algebra 2 / 17
4 The tropical semiring The tropical (or max-plus) semiring has elements R = R { } and binary operations x y = max(x, y); and x y = x + y. Properties R is an idempotent semifield: (R, ) is an abelian group with identity 0; is a zero element for ; (R, ) is a commutative monoid with identity ; distributes over ; x x = x Johnson & Kambites Tropical matrix algebra 2 / 17
5 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Johnson & Kambites Tropical matrix algebra 3 / 17
6 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Applications Tropical methods have applications in... Johnson & Kambites Tropical matrix algebra 3 / 17
7 The tropical semiring Tropical matrix algebra or max-plus algebra is linear algebra where the base field is replaced by the tropical semiring. Applications Tropical methods have applications in... Combinatorial Optimisation Discrete Event Systems Control Theory Formal Languages and Automata Phylogenetics Statistical Inference Geometric Group Theory Enumerative Algebraic Geometry Johnson & Kambites Tropical matrix algebra 3 / 17
8 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Johnson & Kambites Tropical matrix algebra 4 / 17
9 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Question What is its abstract algebraic structure? Johnson & Kambites Tropical matrix algebra 4 / 17
10 Tropical matrices We (hope to) study the semigroup M n (R) of all n n tropical matrices under multiplication. Question What is its abstract algebraic structure? For example, what are its... Ideals? Idempotents? Subgroups? Johnson & Kambites Tropical matrix algebra 4 / 17
11 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Johnson & Kambites Tropical matrix algebra 5 / 17
12 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... Johnson & Kambites Tropical matrix algebra 5 / 17
13 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
14 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,b) = (a,b) + (c,d) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
15 Affine tropical n-space M n (R) comes equipped with a natural action on the space R n of tropical n-vectors (affine tropical n-space). Example We may think of elements of tropical 2-space pictorially as follows... (-,b) (a,b) (c,b) = (a,b) + (c,d) k x (c,b) (c,d) (-,- ) (c,- ) Johnson & Kambites Tropical matrix algebra 5 / 17
16 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Johnson & Kambites Tropical matrix algebra 6 / 17
17 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (-,f) m x (a,b) (a,b) (c,d) k x (c,d) (-,- ) (e,- ) Johnson & Kambites Tropical matrix algebra 6 / 17
18 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (a,b) (-,f) [-,f ] m x (a,b) (c,d) [a,b] [c,d] k x (c,d) (-,- ) (e,- ) [e,- ] Johnson & Kambites Tropical matrix algebra 6 / 17
19 Projective tropical (n 1)-space From R n we obtain projective tropical (n 1)-space by discarding the zero vector and identifying two vectors which are tropical scalings of each other. Example (a,b) (-,f) [-,f ] m x (a,b) (c,d) [a,b] [c,d] b-a k x (c,d) d-c (-,- ) (e,- ) [e,- ] - Johnson & Kambites Tropical matrix algebra 6 / 17
20 Projective tropical 1-space Thus we identify projective tropical 1-space with the two-point compactification of the real line ˆR = R {, } via the map [-,f ] [x, y] y x. [a,b] [c,d] b-a d-c [e,- ] - Johnson & Kambites Tropical matrix algebra 7 / 17
21 Projective tropical 1-space Thus we identify projective tropical 1-space with the two-point compactification of the real line ˆR = R {, } via the map [-,f ] [x, y] y x. [a,b] [c,d] b-a d-c [e,- ] - Question How does the algebraic structure of M n (R) relate to the geometric structure of affine tropical n-space and projective tropical (n 1)-space? Johnson & Kambites Tropical matrix algebra 7 / 17
22 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Johnson & Kambites Tropical matrix algebra 8 / 17
23 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Example ( ) a Let A = M b c 2 (R), where a, b, c R. Johnson & Kambites Tropical matrix algebra 8 / 17
24 Column and row spaces For A M n (R) we write C(A) for the column span of A (a tropical subspace in R n ); R(A) for the row span of A (a tropical subspace in R n ). Example ( ) a Let A = M b c 2 (R), where a, b, c R. Then C(A) = {( x y ) } : x + b a y R 2. b-a Johnson & Kambites Tropical matrix algebra 8 / 17
25 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Johnson & Kambites Tropical matrix algebra 9 / 17
26 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Example In the case n = ( 2, convex ) sets in ˆR are just intervals. a Consider A = M b c 2 (R), where a, b, c R. Johnson & Kambites Tropical matrix algebra 9 / 17
27 Projective column and row spaces For A M n (R) we write PC(A) for the image of C(A) in projective space (a convex set); PR(A) for the image of R(A) in projective space (a convex set). Example In the case n = ( 2, convex ) sets in ˆR are just intervals. a Consider A = M b c 2 (R), where a, b, c R. b-a Then PC(A) = [b a, ] ˆR. Johnson & Kambites Tropical matrix algebra 9 / 17
28 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. Johnson & Kambites Tropical matrix algebra 10 / 17
29 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation xry xm = ym x R y and y R x Johnson & Kambites Tropical matrix algebra 10 / 17
30 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, xly Mx = My xj y MxM = MyM; Johnson & Kambites Tropical matrix algebra 10 / 17
31 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, We also define equivalence relations... xhy xry and xly; xly Mx = My xdy xrz and zly for some z M; xj y MxM = MyM; Johnson & Kambites Tropical matrix algebra 10 / 17
32 Ideals and Green s relations We define a pre-order R on a monoid M by x R y xm ym. From this we obtain an equivalence relation Similarly... xry xm = ym x R y and y R x x L y Mx My, x J y MxM MyM, We also define equivalence relations... Note xhy xry and xly; xly Mx = My xdy xrz and zly for some z M; xj y MxM = MyM; These relations encapsulate the (left, right and two-sided) ideal structure of M and are fundamental to its structure. Johnson & Kambites Tropical matrix algebra 10 / 17
33 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Johnson & Kambites Tropical matrix algebra 11 / 17
34 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Johnson & Kambites Tropical matrix algebra 11 / 17
35 Green s R relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A R B; (ii) C(A) C(B); (iii) PC(A) PC(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). So R-classes in M n (R) are in 1-1 correspondence with n-generated convex sets in projective tropical (n 1)-space. Johnson & Kambites Tropical matrix algebra 11 / 17
36 Green s L relation in M n (R). Lemma Let A, B M n (R). Then the following are equivalent: (i) A L B; (ii) R(A) R(B); (iii) PR(A) PR(B). Corollary Let A, B M n (R). Then the following are equivalent: (i) ALB; (ii) R(A) = R(B); (iii) PR(A) = PR(B). So L-classes in M n (R) are in 1-1 correspondence with n-generated convex sets in projective tropical (n 1)-space. Johnson & Kambites Tropical matrix algebra 12 / 17
37 Green s R relation in M 2 (R). Corollary Let A, B M 2 (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Johnson & Kambites Tropical matrix algebra 13 / 17
38 Green s R relation in M 2 (R). Corollary Let A, B M 2 (R). Then the following are equivalent: (i) ARB; (ii) C(A) = C(B); (iii) PC(A) = PC(B). Corollary The lattice of principal right ideals in M 2 (R) is isomorphic to the intersection lattice generated by closed subintervals of the closed unit interval. Johnson & Kambites Tropical matrix algebra 13 / 17
39 Isometries in projective tropical 1-space We can define a metric on ˆR = R {, } by 0 if x = y d(x, y) = if x = y or x = y y x otherwise. This gives a natural notion of isometry (denoted by =). Johnson & Kambites Tropical matrix algebra 14 / 17
40 Isometries in projective tropical 1-space We can define a metric on ˆR = R {, } by 0 if x = y d(x, y) = if x = y or x = y y x otherwise. This gives a natural notion of isometry (denoted by =). Proposition Let A M 2 (R). Then PC(A) = PR(A). Johnson & Kambites Tropical matrix algebra 14 / 17
41 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Johnson & Kambites Tropical matrix algebra 15 / 17
42 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Theorem Let A, B M 2 (R). Then the following are equivalent (i) AJ B; (ii) ADB; (iii) PC(A) = PC(B) (iv) PR(A) = PR(B) Johnson & Kambites Tropical matrix algebra 15 / 17
43 Green s J relation in M 2 (R) Proposition Let A, B M 2 (R). Then A J B if and only if PC(A) embeds isometrically in PC(B). Theorem Let A, B M 2 (R). Then the following are equivalent (i) AJ B; (ii) ADB; (iii) PC(A) = PC(B) (iv) PR(A) = PR(B) Corollary The lattice of principal two-sided ideals in M 2 (R) is isomorphic to the lattice of isometry types of closed convex subsets of ˆR. Johnson & Kambites Tropical matrix algebra 15 / 17
44 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 ( x + y x y 0 ) and ( ) where x, y R with x + y 0. Johnson & Kambites Tropical matrix algebra 16 / 17
45 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 where x, y R with x + y 0. Fact ( x + y x y 0 ) and ( For every 2-generated convex subset X of ˆR, there is an idempotent E M 2 (R) with PC(E) = X. Thus M 2 (R) is regular. ) Johnson & Kambites Tropical matrix algebra 16 / 17
46 Idempotents and regularity The idempotents in M 2 (R) are ( ) ( ) 0 x 0 x,, y x + y y 0 where x, y R with x + y 0. Fact ( x + y x y 0 ) and ( For every 2-generated convex subset X of ˆR, there is an idempotent E M 2 (R) with PC(E) = X. Thus M 2 (R) is regular. ) Example Consider X = [b a, ] ˆR. Then ( we can choose ) 0 E = M b a 0 2 (R) such that PC(E) = X b-a Johnson & Kambites Tropical matrix algebra 16 / 17
47 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Johnson & Kambites Tropical matrix algebra 17 / 17
48 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Theorem Let M ˆR be a closed convex subset. The maximal subgroups in the D-class corresponding to M are isomorphic to: {1} if M = ; R if M is a point or an interval with one real endpoint; R S 2 if M is an interval with 2 real endpoints; R S 2 if M = ˆR. Johnson & Kambites Tropical matrix algebra 17 / 17
49 Groups of 2 2 tropical matrices Let S be a semigroup. It is well known that the maximal subgroups of S are exactly the H-classes of idempotents and that any two maximal subgroups in the same D-class are isomorphic. Theorem Let M ˆR be a closed convex subset. The maximal subgroups in the D-class corresponding to M are isomorphic to: Corollary {1} if M = ; R if M is a point or an interval with one real endpoint; R S 2 if M is an interval with 2 real endpoints; R S 2 if M = ˆR. Every group of 2 2 tropical matrices is torsion-free abelian, or has a torsion-free abelian subgroup of index 2. Johnson & Kambites Tropical matrix algebra 17 / 17
Multiplicative structure of 2x2 tropical matrices. Johnson, Marianne and Kambites, Mark. MIMS EPrint:
Multiplicative structure of 2x2 tropical matrices Johnson, Marianne and Kambites, Mark 2009 MIMS EPrint: 2009.74 Manchester Institute for Mathematical Sciences School of Mathematics The Universit of Manchester
More informationTROPICAL SCHEME THEORY. Idempotent semirings
TROPICAL SCHEME THEORY Idempotent semirings Definition 0.1. A semiring is (R, +,, 0, 1) such that (R, +, 0) is a commutative monoid (so + is a commutative associative binary operation on R 0 is an additive
More informationOn Linear Subspace Codes Closed under Intersection
On Linear Subspace Codes Closed under Intersection Pranab Basu Navin Kashyap Abstract Subspace codes are subsets of the projective space P q(n), which is the set of all subspaces of the vector space F
More information2. ETALE GROUPOIDS MARK V. LAWSON
2. ETALE GROUPOIDS MARK V. LAWSON Abstract. In this article, we define étale groupoids and describe some of their properties. 1. Generalities 1.1. Categories. A category is usually regarded as a category
More informationIdentities in upper triangular tropical matrix semigroups and the bicyclic monoid
Identities in upper triangular tropical matrix semigroups and the bicyclic monoid Marianne Johnson (University of Manchester) Joint work with Laure Daviaud and Mark Kambites AAA94 + NSAC 2017, 17th June
More informationMonoids and Their Cayley Graphs
Monoids and Their Cayley Graphs Nik Ruskuc nik@mcs.st-and.ac.uk School of Mathematics and Statistics, University of St Andrews NBGGT, Leeds, 30 April, 2008 Instead of an Apology Then, as for the geometrical
More informationPermutation Groups and Transformation Semigroups Lecture 2: Semigroups
Permutation Groups and Transformation Semigroups Lecture 2: Semigroups Peter J. Cameron Permutation Groups summer school, Marienheide 18 22 September 2017 I am assuming that you know what a group is, but
More informationarxiv: v1 [math.ra] 23 Feb 2018
JORDAN DERIVATIONS ON SEMIRINGS OF TRIANGULAR MATRICES arxiv:180208704v1 [mathra] 23 Feb 2018 Abstract Dimitrinka Vladeva University of forestry, bulklohridski 10, Sofia 1000, Bulgaria E-mail: d vladeva@abvbg
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationDISCRETE SUBGROUPS, LATTICES, AND UNITS.
DISCRETE SUBGROUPS, LATTICES, AND UNITS. IAN KIMING 1. Discrete subgroups of real vector spaces and lattices. Definitions: A lattice in a real vector space V of dimension d is a subgroup of form: Zv 1
More informationON UPPER TRIANGULAR TROPICAL MATRIX SEMIGROUPS, TROPICAL MATRIX IDENTITIES AND T-MODULES
ON UPPER TRIANGULAR TROPICAL MATRIX SEMIGROUPS, TROPICAL MATRIX IDENTITIES AND T-MODULES A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science
More informationRegular dilation and Nica-covariant representation on right LCM semigroups
Regular dilation and Nica-covariant representation on right LCM semigroups Boyu Li University of Waterloo COSy, June 2018 University of Manitoba Question Given a contractive representation T of a semigroup
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationTHE SEMIGROUP βs APPLICATIONS TO RAMSEY THEORY
THE SEMIGROUP βs If S is a discrete space, its Stone-Čech compactification βs can be described as the space of ultrafilters on S with the topology for which the sets of the form A = {p βs : A p}, where
More informationSubrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING
Subrings and Ideals Chapter 2 2.1 INTRODUCTION In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ) is a ring and S is a non-empty
More informationMAXIMAL ORDERS IN COMPLETELY 0-SIMPLE SEMIGROUPS
MAXIMAL ORDERS IN COMPLETELY 0-SIMPLE SEMIGROUPS John Fountain and Victoria Gould Department of Mathematics University of York Heslington York YO1 5DD, UK e-mail: jbf1@york.ac.uk varg1@york.ac.uk Abstract
More informationZ n -free groups are CAT(0)
Z n -free groups are CAT(0) Inna Bumagin joint work with Olga Kharlampovich to appear in the Journal of the LMS February 6, 2014 Introduction Lyndon Length Function Let G be a group and let Λ be a totally
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 22. Smith normal form of an integer matrix (linear algebra over Z).
18.312: Algebraic Combinatorics Lionel Levine Lecture date: May 3, 2011 Lecture 22 Notes by: Lou Odette This lecture: Smith normal form of an integer matrix (linear algebra over Z). 1 Review of Abelian
More informationOn Regularity of Incline Matrices
International Journal of Algebra, Vol. 5, 2011, no. 19, 909-924 On Regularity of Incline Matrices A. R. Meenakshi and P. Shakila Banu Department of Mathematics Karpagam University Coimbatore-641 021, India
More informationTropical Optimization Framework for Analytical Hierarchy Process
Tropical Optimization Framework for Analytical Hierarchy Process Nikolai Krivulin 1 Sergeĭ Sergeev 2 1 Faculty of Mathematics and Mechanics Saint Petersburg State University, Russia 2 School of Mathematics
More informationRelations and Equivalence Relations
Relations and Equivalence Relations In this section, we shall introduce a formal definition for the notion of a relation on a set. This is something we often take for granted in elementary algebra courses,
More informationA Solution of a Tropical Linear Vector Equation
A Solution of a Tropical Linear Vector Equation NIKOLAI KRIVULIN Faculty of Mathematics and Mechanics St. Petersburg State University 28 Universitetsky Ave., St. Petersburg, 198504 RUSSIA nkk@math.spbu.ru
More informationFiniteness conditions and index in semigroup theory
Finiteness conditions and index in semigroup theory Robert Gray University of Leeds Leeds, January 2007 Robert Gray (University of Leeds) 1 / 39 Outline 1 Motivation and background Finiteness conditions
More information4.4. Orthogonality. Note. This section is awesome! It is very geometric and shows that much of the geometry of R n holds in Hilbert spaces.
4.4. Orthogonality 1 4.4. Orthogonality Note. This section is awesome! It is very geometric and shows that much of the geometry of R n holds in Hilbert spaces. Definition. Elements x and y of a Hilbert
More informationAN ALGEBRA APPROACH TO TROPICAL MATHEMATICS
Emory University: Saltman Conference AN ALGEBRA APPROACH TO TROPICAL MATHEMATICS Louis Rowen, Department of Mathematics, Bar-Ilan University Ramat-Gan 52900, Israel (Joint work with Zur Izhakian) May,
More informationIdentities in upper triangular tropical matrix semigroups and the bicyclic monoid
Identities in upper triangular tropical matrix semigroups and the bicyclic monoid Marianne Johnson (Joint work with Laure Daviaud and Mark Kambites) York Semigroup Seminar, May 2017 Semigroup identities
More informationA SURVEY OF CLUSTER ALGEBRAS
A SURVEY OF CLUSTER ALGEBRAS MELODY CHAN Abstract. This is a concise expository survey of cluster algebras, introduced by S. Fomin and A. Zelevinsky in their four-part series of foundational papers [1],
More informationLecture 10: Limit groups
Lecture 10: Limit groups Olga Kharlampovich November 4 1 / 16 Groups universally equivalent to F Unification Theorem 1 Let G be a finitely generated group and F G. Then the following conditions are equivalent:
More information1.3 Group Actions. Exercise Prove that a CAT(1) piecewise spherical simplicial complex is metrically flag.
Exercise 1.2.6. Prove that a CAT(1) piecewise spherical simplicial complex is metrically flag. 1.3 Group Actions Definition 1.3.1. Let X be a metric space, and let λ : X X be an isometry. The displacement
More informationALGEBRAIC GROUPS J. WARNER
ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
More information4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;
4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length
More informationBasic Properties of Metric and Normed Spaces
Basic Properties of Metric and Normed Spaces Computational and Metric Geometry Instructor: Yury Makarychev The second part of this course is about metric geometry. We will study metric spaces, low distortion
More informationALGEBRAIC GEOMETRY I, FALL 2016.
ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of
More informationA GENERALIZATION OF BI IDEALS IN SEMIRINGS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 123-133 DOI: 10.7251/BIMVI1801123M Former BULLETIN
More informationLINEAR ALGEBRA: THEORY. Version: August 12,
LINEAR ALGEBRA: THEORY. Version: August 12, 2000 13 2 Basic concepts We will assume that the following concepts are known: Vector, column vector, row vector, transpose. Recall that x is a column vector,
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationDiscrete Mathematics. Benny George K. September 22, 2011
Discrete Mathematics Benny George K Department of Computer Science and Engineering Indian Institute of Technology Guwahati ben@iitg.ernet.in September 22, 2011 Set Theory Elementary Concepts Let A and
More informationCartan MASAs and Exact Sequences of Inverse Semigroups
Cartan MASAs and Exact Sequences of Inverse Semigroups Adam H. Fuller (University of Nebraska - Lincoln) joint work with Allan P. Donsig and David R. Pitts NIFAS Nov. 2014, Des Moines, Iowa Cartan MASAs
More informationLattices, closure operators, and Galois connections.
125 Chapter 5. Lattices, closure operators, and Galois connections. 5.1. Semilattices and lattices. Many of the partially ordered sets P we have seen have a further valuable property: that for any two
More information0.2 Vector spaces. J.A.Beachy 1
J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a
More informationNotes taken by Costis Georgiou revised by Hamed Hatami
CSC414 - Metric Embeddings Lecture 6: Reductions that preserve volumes and distance to affine spaces & Lower bound techniques for distortion when embedding into l Notes taken by Costis Georgiou revised
More informationAlgebraic closure of some generalized convex sets
Algebraic closure of some generalized convex sets Anna B. ROMANOWSKA* Warsaw University of Technology, Poland Gábor CZÉDLI University of Szeged, Hungary 1 AFFINE SPACES AND CONVEX SETS 1. Real affine spaces
More informationP-Spaces and the Prime Spectrum of Commutative Semirings
International Mathematical Forum, 3, 2008, no. 36, 1795-1802 P-Spaces and the Prime Spectrum of Commutative Semirings A. J. Peña Departamento de Matemáticas, Facultad Experimental de Ciencias, Universidad
More informationarxiv: v1 [math.co] 24 Sep 2013
Tropical Cramer Determinants Revisited Marianne Akian, Stéphane Gaubert, and Alexander Guterman arxiv:1309.6298v1 [math.co] 24 Sep 2013 We dedicate this paper to the memory of our friend and colleague
More informationThe Symmetric Space for SL n (R)
The Symmetric Space for SL n (R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SL n (R) acts. Here, as usual, SL n (R) denotes the group of n n
More informationZERO DIVISORS FREE Γ SEMIRING
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 37-43 DOI: 10.7251/BIMVI1801037R Former BULLETIN OF
More informationa (b + c) = a b + a c
Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationWhen does a semiring become a residuated lattice?
When does a semiring become a residuated lattice? Ivan Chajda and Helmut Länger arxiv:1809.07646v1 [math.ra] 20 Sep 2018 Abstract It is an easy observation that every residuated lattice is in fact a semiring
More informationALGEBRAIC PROPERTIES OF THE LATTICE R nx
CHAPTER 6 ALGEBRAIC PROPERTIES OF THE LATTICE R nx 6.0 Introduction This chapter continues to explore further algebraic and topological properties of the complete lattice Rnx described in Chapter 5. We
More informationarxiv:math/ v1 [math.gr] 1 Jan 1992
APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 1, Jan 1992, Pages 87-112 Λ-TREES AND THEIR APPLICATIONS arxiv:math/9201265v1 [math.gr] 1 Jan 1992 John W. Morgan To most mathematicians
More informationUnions of Dominant Chains of Pairwise Disjoint, Completely Isolated Subsemigroups
Palestine Journal of Mathematics Vol. 4 (Spec. 1) (2015), 490 495 Palestine Polytechnic University-PPU 2015 Unions of Dominant Chains of Pairwise Disjoint, Completely Isolated Subsemigroups Karen A. Linton
More informationIn this paper we introduce min-plus low rank
arxiv:78.655v [math.na] Aug 7 Min-plus algebraic low rank matrix approximation: a new method for revealing structure in networks James Hook University of Bath, United Kingdom In this paper we introduce
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular p-chains with coefficients in a field K. Furthermore, we can define the
More informationValence automata over E-unitary inverse semigroups
Valence automata over E-unitary inverse semigroups Erzsi Dombi 30 May 2018 Outline Motivation Notation and introduction Valence automata Bicyclic and polycyclic monoids Motivation Chomsky-Schützenberger
More informationreplacements Expansions of Semigroups Jon McCammond U.C. Santa Barbara
replacements Epansions of Semigroups 1 1 Jon McCammond.C. Santa Barbara 1 Main theorem Rough version Thm(M-Rhodes) If S is a finite A-semigroup then there eists a finite epansion of S such that the right
More informationSemigroup, monoid and group models of groupoid identities. 1. Introduction
Quasigroups and Related Systems 16 (2008), 25 29 Semigroup, monoid and group models of groupoid identities Nick C. Fiala Abstract In this note, we characterize those groupoid identities that have a (nite)
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.
MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationArkadiusz Męcel. Subspace semigroups of finite dimensional algebras. Uniwersytet Warszawski. Praca semestralna nr 3 (semestr letni 2011/12)
Arkadiusz Męcel Uniwersytet Warszawski Subspace semigroups of finite dimensional algebras Praca semestralna nr 3 (semestr letni 2011/12) Opiekun pracy: Grzegorz Bobiński Nicolaus Copernicus University
More informationRING ELEMENTS AS SUMS OF UNITS
1 RING ELEMENTS AS SUMS OF UNITS CHARLES LANSKI AND ATTILA MARÓTI Abstract. In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand
More informationTightness and Inverse Semigroups
Tightness and Inverse Semigroups Allan Donsig University of Nebraska Lincoln April 14, 2012 This work is a) joint with David Milan, and b) in progress. Allan Donsig (UNL) Tightness and Inverse Semigroups
More informationON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #A21 ON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS Sergey Kitaev Department of Mathematics, University of Kentucky,
More informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
More informationA Weak Bisimulation for Weighted Automata
Weak Bisimulation for Weighted utomata Peter Kemper College of William and Mary Weighted utomata and Semirings here focus on commutative & idempotent semirings Weak Bisimulation Composition operators Congruence
More informationThe (extended) rank weight enumerator and q-matroids
The (extended) rank weight enumerator and q-matroids Relinde Jurrius Ruud Pellikaan Vrije Universiteit Brussel, Belgium Eindhoven University of Technology, The Netherlands Conference on Random network
More informationarxiv: v2 [math.gr] 25 Mar 2013
MAXIMAL SUBGROUPS OF FREE IDEMPOTENT GENERATED SEMIGROUPS OVER THE FULL LINEAR MONOID arxiv:1112.0893v2 [math.gr] 25 Mar 2013 IGOR DOLINKA AND ROBERT D. GRAY Abstract. We show that the rank r component
More informationOn finite congruence-simple semirings
On finite congruence-simple semirings arxiv:math/0205083v2 [math.ra] 19 Aug 2003 Chris Monico Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409-1042 cmonico@math.ttu.edu
More informationCourse of algebra. Introduction and Problems
Course of algebra. Introduction and Problems A.A.Kirillov Fall 2008 1 Introduction 1.1 What is Algebra? Answer: Study of algebraic operations. Algebraic operation: a map M M M. Examples: 1. Addition +,
More informationTactical Decompositions of Steiner Systems and Orbits of Projective Groups
Journal of Algebraic Combinatorics 12 (2000), 123 130 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Tactical Decompositions of Steiner Systems and Orbits of Projective Groups KELDON
More informationREGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
More informationA Tropical Extremal Problem with Nonlinear Objective Function and Linear Inequality Constraints
A Tropical Extremal Problem with Nonlinear Objective Function and Linear Inequality Constraints NIKOLAI KRIVULIN Faculty of Mathematics and Mechanics St. Petersburg State University 28 Universitetsky Ave.,
More informationIntroduction to Association Schemes
Introduction to Association Schemes Akihiro Munemasa Tohoku University June 5 6, 24 Algebraic Combinatorics Summer School, Sendai Assumed results (i) Vandermonde determinant: a a m =. a m a m m i
More informationAbstract Convexity: Results and Speculations
Abstract Convexity: Results and Speculations Tobias Fritz Max Planck Institut für Mathematik, Bonn Mathematical Colloquium Hagen, 26 Nov 29 Overview. Introducing convex spaces 2. A Hahn-Banach theorem
More informationCS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a
Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations
More informationMATH 433 Applied Algebra Lecture 22: Semigroups. Rings.
MATH 433 Applied Algebra Lecture 22: Semigroups. Rings. Groups Definition. A group is a set G, together with a binary operation, that satisfies the following axioms: (G1: closure) for all elements g and
More informationClC (X ) : X ω X } C. (11)
With each closed-set system we associate a closure operation. Definition 1.20. Let A, C be a closed-set system. Define Cl C : : P(A) P(A) as follows. For every X A, Cl C (X) = { C C : X C }. Cl C (X) is
More informationA Grassmann Algebra for Matroids
Joint work with Jeffrey Giansiracusa, Swansea University, U.K. Matroid (Whitney, 1935) A collection B of subsets of [n] = {1,2,...,n}, called bases, such that the basis exchange property holds: Matroid
More informationA four element semigroup that is inherently nonfinitely based?
A four element semigroup that is inherently nonfinitely based? Peter R. Jones Marquette University NBSAN Workshop, York, November 21, 2012 Finite basis properties A (finite) algebra A is finitely based
More informationGEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS
GEOMETRIC CONSTRUCTIONS AND ALGEBRAIC FIELD EXTENSIONS JENNY WANG Abstract. In this paper, we study field extensions obtained by polynomial rings and maximal ideals in order to determine whether solutions
More informationFormal groups. Peter Bruin 2 March 2006
Formal groups Peter Bruin 2 March 2006 0. Introduction The topic of formal groups becomes important when we want to deal with reduction of elliptic curves. Let R be a discrete valuation ring with field
More informationL fuzzy ideals in Γ semiring. M. Murali Krishna Rao, B. Vekateswarlu
Annals of Fuzzy Mathematics and Informatics Volume 10, No. 1, (July 2015), pp. 1 16 ISSN: 2093 9310 (print version) ISSN: 2287 6235 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com
More informationEIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND
EIGENVECTORS FOR A RANDOM WALK ON A LEFT-REGULAR BAND FRANCO SALIOLA Abstract. We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups
More informationA Multiplicative Operation on Matrices with Entries in an Arbitrary Abelian Group
A Multiplicative Operation on Matrices with Entries in an Arbitrary Abelian Group Cyrus Hettle (cyrus.h@uky.edu) Robert P. Schneider (robert.schneider@uky.edu) University of Kentucky Abstract We define
More informationRow Space, Column Space, and Nullspace
Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space
More informationTropical decomposition of symmetric tensors
Tropical decomposition of symmetric tensors Melody Chan University of California, Berkeley mtchan@math.berkeley.edu December 11, 008 1 Introduction In [], Comon et al. give an algorithm for decomposing
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationChapter 3. Introducing Groups
Chapter 3 Introducing Groups We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs
More informationSOLUTION OF GENERALIZED LINEAR VECTOR EQUATIONS IN IDEMPOTENT ALGEBRA
, pp. 23 36, 2006 Vestnik S.-Peterburgskogo Universiteta. Matematika UDC 519.63 SOLUTION OF GENERALIZED LINEAR VECTOR EQUATIONS IN IDEMPOTENT ALGEBRA N. K. Krivulin The problem on the solutions of homogeneous
More informationResearch Statement. MUHAMMAD INAM 1 of 5
MUHAMMAD INAM 1 of 5 Research Statement Preliminaries My primary research interests are in geometric inverse semigroup theory and its connections with other fields of mathematics. A semigroup M is called
More informationAn Algebraic Approach to Energy Problems I -Continuous Kleene ω-algebras
Acta Cybernetica 23 (2017) 203 228. An Algebraic Approach to Energy Problems I -Continuous Kleene ω-algebras Zoltán Ésika, Uli Fahrenberg b, Axel Legay c, and Karin Quaas d Abstract Energy problems are
More information#A63 INTEGERS 17 (2017) CONCERNING PARTITION REGULAR MATRICES
#A63 INTEGERS 17 (2017) CONCERNING PARTITION REGULAR MATRICES Sourav Kanti Patra 1 Department of Mathematics, Ramakrishna Mission Vidyamandira, Belur Math, Howrah, West Bengal, India souravkantipatra@gmail.com
More informationCombinatorics for algebraic geometers
Combinatorics for algebraic geometers Calculations in enumerative geometry Maria Monks March 17, 214 Motivation Enumerative geometry In the late 18 s, Hermann Schubert investigated problems in what is
More informationReconstruction and Higher Dimensional Geometry
Reconstruction and Higher Dimensional Geometry Hongyu He Department of Mathematics Louisiana State University email: hongyu@math.lsu.edu Abstract Tutte proved that, if two graphs, both with more than two
More informationLeft almost semigroups dened by a free algebra. 1. Introduction
Quasigroups and Related Systems 16 (2008), 69 76 Left almost semigroups dened by a free algebra Qaiser Mushtaq and Muhammad Inam Abstract We have constructed LA-semigroups through a free algebra, and the
More informationGROUPS DEFINABLE IN O-MINIMAL STRUCTURES
GROUPS DEFINABLE IN O-MINIMAL STRUCTURES PANTELIS E. ELEFTHERIOU Abstract. In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups definable
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More information