generalized bol moufang loop varieties not just for
play

Generalized Bol-Moufang loop varietiesnot just for breakfast! J.D. - PowerPoint PPT Presentation

Apr 02, 2023 •629 likes •1.29k views

Generalized Bol-Moufang loop varietiesnot just for breakfast! J.D. Phillips Northern Michigan University Loops 19, Budapest, 11 July 2019 Loops 19, Budapest, 11 July 2019 1 / 15 The Six Instigator Identities Loops 19, Budapest,

  1. Generalized Bol-Moufang loop varieties—not just for breakfast! J.D. Phillips Northern Michigan University Loops ’19, Budapest, 11 July 2019 Loops ’19, Budapest, 11 July 2019 1 / 15

  2. The Six Instigator Identities Loops ’19, Budapest, 11 July 2019 2 / 15

  3. The Six Instigator Identities z ( y · zx ) = ( z · yz ) x Loops ’19, Budapest, 11 July 2019 2 / 15

  4. The Six Instigator Identities z ( y · zx ) = ( z · yz ) x x ( zy · z ) = ( xz · y ) z Loops ’19, Budapest, 11 July 2019 2 / 15

  5. The Six Instigator Identities z ( y · zx ) = ( z · yz ) x x ( zy · z ) = ( xz · y ) z z ( x · zy ) = ( zx · z ) y ( xz · y ) z = x ( z · yz ) ( z · xy ) z = zx · yz z ( xy · z ) = zx · yz Loops ’19, Budapest, 11 July 2019 2 / 15

  6. Commonalities Loops ’19, Budapest, 11 July 2019 3 / 15

  7. Commonalities 1. contain only one operation—the loop product, Loops ’19, Budapest, 11 July 2019 3 / 15

  8. Commonalities 1. contain only one operation—the loop product, 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign, and Loops ’19, Budapest, 11 July 2019 3 / 15

  9. Commonalities 1. contain only one operation—the loop product, 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign, and 3. the order in which the variables appear is the same on each side of the equal sign. Loops ’19, Budapest, 11 July 2019 3 / 15

  10. Commonalities 1. contain only one operation—the loop product, 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign, and 3. the order in which the variables appear is the same on each side of the equal sign. Bol-Moufang type Loops ’19, Budapest, 11 July 2019 3 / 15

  11. Algebraic Setting Loops ’19, Budapest, 11 July 2019 4 / 15

  12. Algebraic Setting 60 such identities. Loops ’19, Budapest, 11 July 2019 4 / 15

  13. Algebraic Setting 60 such identities. Bol-Moufang loop variety . Loops ’19, Budapest, 11 July 2019 4 / 15

  14. Algebraic Setting 60 such identities. Bol-Moufang loop variety . There are 14 such loop varieties. Loops ’19, Budapest, 11 July 2019 4 / 15

  15. Algebraic Setting 60 such identities. Bol-Moufang loop variety . There are 14 such loop varieties. Ferenc Fenyves. Loops ’19, Budapest, 11 July 2019 4 / 15

  16. A bit more generally Loops ’19, Budapest, 11 July 2019 5 / 15

  17. A bit more generally CML: xx · yz = xy · xz Loops ’19, Budapest, 11 July 2019 5 / 15

  18. A bit more generally CML: xx · yz = xy · xz Left Cheban: x ( xy · z ) = yx · xz Loops ’19, Budapest, 11 July 2019 5 / 15

  19. A bit more generally CML: xx · yz = xy · xz Left Cheban: x ( xy · z ) = yx · xz Right Cheban: zx · xy = ( z · yx ) x Loops ’19, Budapest, 11 July 2019 5 / 15

  20. A bit more generally CML: xx · yz = xy · xz Left Cheban: x ( xy · z ) = yx · xz Right Cheban: zx · xy = ( z · yx ) x Cheban: x ( xy · z ) = ( y · zx ) x Loops ’19, Budapest, 11 July 2019 5 / 15

  21. Commonalities Loops ’19, Budapest, 11 July 2019 6 / 15

  22. Commonalities 1. contain only one operation—the loop product, and 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. Loops ’19, Budapest, 11 July 2019 6 / 15

  23. Commonalities 1. contain only one operation—the loop product, and 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. Generalized Bol-Moufang type . Loops ’19, Budapest, 11 July 2019 6 / 15

  24. Commonalities 1. contain only one operation—the loop product, and 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. Generalized Bol-Moufang type . 1215 such identities. Loops ’19, Budapest, 11 July 2019 6 / 15

  25. Commonalities 1. contain only one operation—the loop product, and 2. exactly three distinct variables appear on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. Generalized Bol-Moufang type . 1215 such identities. Generalized Bol-Moufang loop variety . Loops ’19, Budapest, 11 July 2019 6 / 15

  26. Mea Culpa Loops ’19, Budapest, 11 July 2019 7 / 15

  27. Mea Culpa Loops ’19, Budapest, 11 July 2019 7 / 15

  28. Varieties Loops ’19, Budapest, 11 July 2019 8 / 15

  29. Varieties Generalized Bol-Moufang type : 1. contains only one operation—the loop product, 2. exactly three distinct variables appearing on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. 1215 such identities. Generalized Bol-Moufang loop variety. Loops ’19, Budapest, 11 July 2019 8 / 15

  30. Varieties Generalized Bol-Moufang type : 1. contains only one operation—the loop product, 2. exactly three distinct variables appearing on each side of the equal sign, one appearing twice on each side of the equal sign, the other two appearing once each on each side of the equal sign. 1215 such identities. Generalized Bol-Moufang loop variety. There are 48 such varieties of loops. Loops ’19, Budapest, 11 July 2019 8 / 15

  31. Winnow it Down Loops ’19, Budapest, 11 July 2019 9 / 15

  32. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  33. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  34. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain. Subtract the three Cheban varieties: 25 remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  35. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain. Subtract the three Cheban varieties: 25 remain. Subtract the one associative variety: 24 remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  36. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain. Subtract the three Cheban varieties: 25 remain. Subtract the one associative variety: 24 remain. Subtract the six that can be described using only two variables, e.g., x · yx = y · xx : 18 remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  37. Winnow it Down Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain. Subtract the three Cheban varieties: 25 remain. Subtract the one associative variety: 24 remain. Subtract the six that can be described using only two variables, e.g., x · yx = y · xx : 18 remain. Subtract the 15 varieties that have “very little” structure (e.g., flexible or alternative): 3 remain. Loops ’19, Budapest, 11 July 2019 9 / 15

  38. FRUTE loops Loops ’19, Budapest, 11 July 2019 10 / 15

  39. FRUTE loops ( x · xy ) z = ( y · zx ) x Loops ’19, Budapest, 11 July 2019 10 / 15

  40. FRUTE loops ( x · xy ) z = ( y · zx ) x FRUTE loops are Moufang. Loops ’19, Budapest, 11 July 2019 10 / 15

  41. FRUTE loops ( x · xy ) z = ( y · zx ) x FRUTE loops are Moufang. This variety is one of four generalized Bol-Moufang loop varieties that consist entirely of not necessarily associative, Moufang loops. Loops ’19, Budapest, 11 July 2019 10 / 15

  42. FRUTE loops ( x · xy ) z = ( y · zx ) x FRUTE loops are Moufang. This variety is one of four generalized Bol-Moufang loop varieties that consist entirely of not necessarily associative, Moufang loops. (The other three are: Moufang loops, commutative Moufang loops, extra loops.) Loops ’19, Budapest, 11 July 2019 10 / 15

  43. FRUTE loops ( x · xy ) z = ( y · zx ) x FRUTE loops are Moufang. This variety is one of four generalized Bol-Moufang loop varieties that consist entirely of not necessarily associative, Moufang loops. (The other three are: Moufang loops, commutative Moufang loops, extra loops.) There are no others. Loops ’19, Budapest, 11 July 2019 10 / 15

  44. FRUTE loops ( x · xy ) z = ( y · zx ) x FRUTE loops are Moufang. This variety is one of four generalized Bol-Moufang loop varieties that consist entirely of not necessarily associative, Moufang loops. (The other three are: Moufang loops, commutative Moufang loops, extra loops.) There are no others. FRUTE loops; the final (Moufang) frontier! Loops ’19, Budapest, 11 July 2019 10 / 15

  45. FRUTE loop properties Loops ’19, Budapest, 11 July 2019 11 / 15

  46. FRUTE loop properties More properties of FRUTE loops: Loops ’19, Budapest, 11 July 2019 11 / 15

Recommend

From conormal varieties of Schubert varieties to loop models A. Knutson & P. Zinn-Justin

From conormal varieties of Schubert varieties to loop models A. Knutson & P. Zinn-Justin

From conormal varieties of Schubert varieties to loop models A. Knutson & P. Zinn-Justin LPTHE (UPMC Paris 6), CNRS P. Zinn-Justin From conormal varieties of Schubert varieties to loop models 1 / 29 Introduction Ten years ago, P. Di

1.26k views • 98 slides
A Generalized Brezing-Weng Algorithm for Constructing Pairing-Friendly Ordinary Abelian Varieties

A Generalized Brezing-Weng Algorithm for Constructing Pairing-Friendly Ordinary Abelian Varieties

Abelian Varieties and Pairing-Based Cryptography Constructing Pairing-Friendly Abelian Varieties Analyzing the Algorithm A Generalized Brezing-Weng Algorithm for Constructing Pairing-Friendly Ordinary Abelian Varieties David Freeman Stanford

537 views • 16 slides
Analytic Moufang loops and Malcev algebras. L.Sabinina UAEM, Cuernavaca Groups and Topological

Analytic Moufang loops and Malcev algebras. L.Sabinina UAEM, Cuernavaca Groups and Topological

Analytic Moufang loops and Malcev algebras. L.Sabinina UAEM, Cuernavaca Groups and Topological Groups TRENTO June 17, 2017 Loops and Groups Given a loop ( L , , /, \ , 1). left multiplication L a x = ax right multiplication R b y = yb By

376 views • 15 slides
Units in quasigroups with Bol-Moufang type identities Natalia N. Didurik and Victor A.

Units in quasigroups with Bol-Moufang type identities Natalia N. Didurik and Victor A.

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Units in quasigroups with Bol-Moufang type identities Natalia N. Didurik and Victor A. Shcherbacov

472 views • 34 slides
Isomorphism type of Schubert varieties Ed Richmond 1 William Slofstra 2 1 Oklahoma State University

Isomorphism type of Schubert varieties Ed Richmond 1 William Slofstra 2 1 Oklahoma State University

Isomorphism type of Schubert varieties Ed Richmond 1 William Slofstra 2 1 Oklahoma State University 2 University of Waterloo June 4th, 2018 Isomorphism type of Schubert varieties W Slofstra Generalized Cartan matrices A GCM is an n n matrix A

557 views • 16 slides
Coarse-Grained Parallelism Variable Privatization, Loop Alignment, Loop Fusion, Loop

Coarse-Grained Parallelism Variable Privatization, Loop Alignment, Loop Fusion, Loop

Coarse-Grained Parallelism Variable Privatization, Loop Alignment, Loop Fusion, Loop interchange and skewing, Loop Strip-mining cs6363 1 Introduction Our previous loop transformations target vector and superscalar architectures Now

538 views • 32 slides
Repetition Types of Loops Counting loop Know how many times to loop

Repetition Types of Loops Counting loop Know how many times to loop

Repetition Types of Loops Counting loop Know how many times to loop Sentinel-controlled loop Expect specific input value to end loop Endfile-controlled loop End of data file is end of loop Input validation loop

257 views • 8 slides
Trading Strategies Introduction Trading Loop Trading Loop Trading Loop Trading Loop Three

Trading Strategies Introduction Trading Loop Trading Loop Trading Loop Trading Loop Three

Trading Strategies Introduction Trading Loop Trading Loop Trading Loop Trading Loop Three Strategies 1. Mean Reversion 2. Momentum 3. Pairs Trading Budish, E., Cramton, P ., & Shim, J. (2015). The high-frequency trading arms race:

803 views • 22 slides
1 Checking Legality in Kelly & Pugh Framework Loop Fusion Example (cont) For each dependence,

1 Checking Legality in Kelly & Pugh Framework Loop Fusion Example (cont) For each dependence,

Loop Transformations for Parallelism & Locality Loop Fusion Example Last time What are the dependences? Unimodular transformation framework do i = 1,n What are the dependences? Loop permutation s 1 A(i) = B(i) + 1 Loop

201 views • 3 slides
Introduction to rational points Varieties An open problem Affine varieties Projective varieties

Introduction to rational points Varieties An open problem Affine varieties Projective varieties

Introduction to rational points Bjorn Poonen Introduction to rational points Varieties An open problem Affine varieties Projective varieties Guiding problems Dimension etc. Bjorn Poonen Curves Genus Classification University of

320 views • 19 slides
Mod p points on Shimura varieties Mark Kisin Harvard Review of Shimura varieties: Review of

Mod p points on Shimura varieties Mark Kisin Harvard Review of Shimura varieties: Review of

Mod p points on Shimura varieties Mark Kisin Harvard Review of Shimura varieties: Review of Shimura varieties: Let G be a connected reductive group over Q and X a conjugacy class of maps of algebraic groups over R h : S = Res C / R G m G R .

1.86k views • 105 slides
Loop Optimizations Important because lots of execution Loop Optimizations Loop Optimizations

Loop Optimizations Important because lots of execution Loop Optimizations Loop Optimizations

Loop Optimizations Important because lots of execution Loop Optimizations Loop Optimizations time occurs in loops First, we will identify loops We will study three optimizations Loop-invariant code motion This lecture is

249 views • 5 slides
B A Y C P L E I T S C U PART II univalence n loop e w ? p a t h s loop l o

B A Y C P L E I T S C U PART II univalence n loop e w ? p a t h s loop l o

B A Y C P L E I T S C U PART II univalence n loop e w ? p a t h s loop l o o p A By Jonah Kan - Own work, CC BY-SA 3.0, h ps://commons.wikimedia.org/w/index.php?curid=27584059 I know how to guarantee a combinatorial

571 views • 22 slides
New examples of defective secant varieties of Segre-Veronese varieties (joint work with M. C.

New examples of defective secant varieties of Segre-Veronese varieties (joint work with M. C.

New examples of defective secant varieties of Segre-Veronese varieties (joint work with M. C. Brambilla) Hirotachi Abo Department of Mathematics University of Idaho abo@uidaho.edu http://www.uidaho.edu/abo October 15, 2011 Notation V

408 views • 15 slides
2-loop 1-loop 0.4 0.3 0.2 V I N C I A R O O T 0.1 0 0 0.5 1 1.5 2 log (Q/GeV) 10

2-loop 1-loop 0.4 0.3 0.2 V I N C I A R O O T 0.1 0 0 0.5 1 1.5 2 log (Q/GeV) 10

0.6 (Q) (Q) s s 0.5 2-loop 1-loop 0.4 0.3 0.2 V I N C I A R O O T 0.1 0 0 0.5 1 1.5 2 log (Q/GeV) 10 Skands, TASI Lectures, arXiv:1207.2389 Hartgring, Laenen, Skands, arXiv:1303.4974 Z 3

238 views • 9 slides
Spherical and toroidal Schubert Varieties Reuven Hodges (joint with V. Lakshmibai, M.B. Can)

Spherical and toroidal Schubert Varieties Reuven Hodges (joint with V. Lakshmibai, M.B. Can)

Spherical and toroidal Schubert Varieties Reuven Hodges (joint with V. Lakshmibai, M.B. Can) University of Illinois at Urbana-Champaign AMS Special Session on Combinatorial Lie Theory November 2019 Spherical varieties Spherical varieties G

325 views • 32 slides
Loop Transformations for Parallelism & Locality Previously Loop transformations,

Loop Transformations for Parallelism & Locality Previously Loop transformations,

Loop Transformations for Parallelism & Locality Previously Loop transformations, unimodular transformation framework Loop interchange/permutation Loop reversal Checking transformation legality Today

211 views • 9 slides
On supersingular varieties Ichiro Shimada Hiroshima University 24 September, 2010, Nagoya 1 /

On supersingular varieties Ichiro Shimada Hiroshima University 24 September, 2010, Nagoya 1 /

On supersingular varieties On supersingular varieties Ichiro Shimada Hiroshima University 24 September, 2010, Nagoya 1 / 28 On supersingular varieties Frobenius supersingular varieties Definition Let X be a smooth projective variety over F q

601 views • 28 slides
The Classification of Generalized Riemann Derivatives Stefan Catoiu DePaul University, Chicago

The Classification of Generalized Riemann Derivatives Stefan Catoiu DePaul University, Chicago

Definition and basic properties Generalized vs. Ordinary Differentiation The Classification of Real Generalized Riemann Derivatives The Classification of Complex Generalized Riemann Derivatives The Classification of Generalized Riemann

768 views • 48 slides
Loop Invariants: Part 2 7 January 2019 OSU CSE 1 Maintaining the Loop Invariant A claimed

Loop Invariants: Part 2 7 January 2019 OSU CSE 1 Maintaining the Loop Invariant A claimed

Loop Invariants: Part 2 7 January 2019 OSU CSE 1 Maintaining the Loop Invariant A claimed loop invariant is valid only if the loop body actually maintains the property, i.e., the loop invariant remains true at the end of each execution of

386 views • 20 slides
Loop Statements & Vectorizing Code Chapter 5 Attaway MATLAB 4E for loop used as a

Loop Statements & Vectorizing Code Chapter 5 Attaway MATLAB 4E for loop used as a

Loop Statements & Vectorizing Code Chapter 5 Attaway MATLAB 4E for loop used as a counted loop repeats an action a specified number of times an iterator or loop variable specifies how many times to repeat the action general

745 views • 31 slides
New and Noteworthy Citrus Varieties Citrus species & Citrus Relatives Hundreds of varieties

New and Noteworthy Citrus Varieties Citrus species & Citrus Relatives Hundreds of varieties

New and Noteworthy Citrus Varieties Citrus species & Citrus Relatives Hundreds of varieties available. CITRON Citrus medica The citron is believed to be one of the original kinds of citrus. Trees are small and shrubby with an

760 views • 61 slides
c } false loop body P (postcondition) Loop Invariant Defn : A boolean condition that

c } false loop body P (postcondition) Loop Invariant Defn : A boolean condition that

while (c) { loop body true c } false loop body P (postcondition) Loop Invariant Defn : A boolean condition that is checked immediately before every evaluation of the loop guard . while (c) I //@loop_invariant I; true c { loop

362 views • 8 slides
Trace while Loop, cont. Trace while Loop, cont. Print Welcome to Java Print Welcome to Java int

Trace while Loop, cont. Trace while Loop, cont. Print Welcome to Java Print Welcome to Java int

while Loop Flow Chart int count = 0; while (loop-continuation-condition) { while (count < 100) { // loop-body; Chapter 4 Loops System.out.println("Welcome to Java!"); Statement(s); count++; } } count = 0; Loop false false

215 views • 9 slides

More recommend