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

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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,

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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

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SLIDE 2

The Six Instigator Identities

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The Six Instigator Identities z(y · zx) = (z · yz)x

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The Six Instigator Identities z(y · zx) = (z · yz)x x(zy · z) = (xz · y)z

Loops ’19, Budapest, 11 July 2019 2 / 15

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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

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Commonalities

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SLIDE 7

Commonalities

  • 1. contain only one operation—the loop product,

Loops ’19, Budapest, 11 July 2019 3 / 15

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SLIDE 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

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SLIDE 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

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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

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Algebraic Setting

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Algebraic Setting 60 such identities.

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Algebraic Setting 60 such identities. Bol-Moufang loop variety.

Loops ’19, Budapest, 11 July 2019 4 / 15

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Algebraic Setting 60 such identities. Bol-Moufang loop variety. There are 14 such loop varieties.

Loops ’19, Budapest, 11 July 2019 4 / 15

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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

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A bit more generally

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A bit more generally CML: xx · yz = xy · xz

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A bit more generally CML: xx · yz = xy · xz Left Cheban: x(xy · z) = yx · xz

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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

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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

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Commonalities

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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

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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

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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

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SLIDE 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.

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Mea Culpa

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Mea Culpa

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Varieties

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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

  • n 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

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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

  • n 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

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Winnow it Down

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Winnow it Down

Subtract the 14 B-M varieties: 34 varieties remain.

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Winnow it Down

Subtract the 14 B-M varieties: 34 varieties remain. Subtract the six which are varieties of commutative loops: 28 remain.

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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.

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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.

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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

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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.

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FRUTE loops

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FRUTE loops (x · xy)z = (y · zx)x

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FRUTE loops (x · xy)z = (y · zx)x FRUTE loops are Moufang.

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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

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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

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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

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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!

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FRUTE loop properties

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FRUTE loop properties More properties of FRUTE loops:

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FRUTE loop properties More properties of FRUTE loops:

◮ Automorphic

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FRUTE loop properties More properties of FRUTE loops:

◮ Automorphic ◮ Their squares are in the commutant

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FRUTE loop properties More properties of FRUTE loops:

◮ Automorphic ◮ Their squares are in the commutant ◮ Their cubes are nuclear

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FRUTE loop properties More properties of FRUTE loops:

◮ Automorphic ◮ Their squares are in the commutant ◮ Their cubes are nuclear ◮ Conjugation is automorphic

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FRUTE loop properties More properties of FRUTE loops:

◮ Automorphic ◮ Their squares are in the commutant ◮ Their cubes are nuclear ◮ Conjugation is automorphic ◮ Their derived subloop is a Boolean group

contained in the loop’s center

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Structure theorem

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Structure theorem

Theorem (Drapal, P., ’19)

Locally finite FRUTE loops are precisely the products O × H, where O is a commutative Moufang loop in which all elements are of odd

  • rder, and H is a 2-group such that the derived

subloop H′ is of exponent two and H′ ≤ Z(H).

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Left and Right C-loops

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Left and Right C-loops LC: xx · yz = (x · xy)z

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Left and Right C-loops LC: xx · yz = (x · xy)z RC: x(yz · z) = xy · zz

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Left and Right C-loops LC: xx · yz = (x · xy)z RC: x(yz · z) = xy · zz C: x(y · yz) = (xy · y)z

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Left and Right C-loops LC: xx · yz = (x · xy)z RC: x(yz · z) = xy · zz C: x(y · yz) = (xy · y)z

Theorem (Vojtˇ echovsk´ y, P. ’06)

C-loops have nuclear squares. Moreover, the modulus of a C-loop by its nucleus is a Steiner loop.

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The remaining two varieties

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The remaining two varieties C-loops with central squares have “transparent” extensions.

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The remaining two varieties C-loops with central squares have “transparent” extensions. (x · xy)z = (yz · x)x (subvariety of LC)

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The remaining two varieties C-loops with central squares have “transparent” extensions. (x · xy)z = (yz · x)x (subvariety of LC) x(x · zy) = z(yx · x) (subvariety of RC)

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The remaining two varieties C-loops with central squares have “transparent” extensions. (x · xy)z = (yz · x)x (subvariety of LC) x(x · zy) = z(yx · x) (subvariety of RC)

Theorem

Loops in this variety have central squares. Moreover, loops in this variety modulo their centers are Steiner loops.

Loops ’19, Budapest, 11 July 2019 14 / 15

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K¨

  • sc!

Thanks for your kind attention!

Loops ’19, Budapest, 11 July 2019 15 / 15