MAGMA!

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MAGMA!
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What is a magma?

• A magma (or groupoid, as you may remember it) is
  a set M equipped with a single binary operation
• M x M ? M.
• The operation is closed by definition.

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Examples of Magmi

• First of all, any set with a closed binary
  operation is a magma

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A commutative magma that is not associative
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A commutative magma that is not associative

• Rock, paper, scissors!!

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A commutative magma that is not associative

• Rock, paper, scissors!!
• rp pr p
• rs sr r
• ps sp s
• rr r, pp p, ss s

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But

• r(ps) ? (rp)s

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Just for fun

• How would we introduce and identity??

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Just for fun

• How would we introduce and identity??
• What would we call this move??

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Constructing the Group
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Constructing the Group
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Constructing the Group
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Constructing the Group
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Constructing the Group
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Constructing the Group
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Constructing the Group
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This is not the only way to make a group!!!!
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Behold
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Behold

• THE QUASIGROUP

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Behold

• THE QUASIGROUP
• A magma, s.t. for each a and b in M, there exist
  elements x and y in M s.t.
• ax b, and
• ya b.

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• Because we do not have a commutative property,
  these elements x and y may differ.

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Example of a Quasigroup
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Example of a Quasigroup

• The integers with subtraction

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Example of a Quasigroup

• The integers with subtraction
• The nonzero rationals (or reals) with division

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Alternative definition

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Alternative definition

• A quasigroup is a magma whose
  multiplication table is a latin square

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Behold
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Behold

• THE LOOP

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Behold

• THE LOOP
• A quasigroup with identity e s.t.
• xe ex x

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Behold

• THE LOOP
• A quasigroup with identity e s.t.
• xe ex x
• Examples of loops that are not groups are
  relatively complex and hard to find

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• Why were our quasigroup examples not loops??

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Now, lets try this again
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Constructing the Group
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Constructing the Group
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Constructing the Group
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Constructing the Group
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So

• What is missing from the Loop in order for it to
  turn into a group??

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So

• What is missing from the Loop in order for it to
  turn into a group??
• Associativity, and
• Unique inverses

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• With the addition of the identity element, the
  divisibility property gives us left and right
  inverses, but they may not be the same.

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It turns out

• that with associativity, combined with the
  identity element of the loop and the divisibility
  of the quasigroup, the inverse of each element is
  unique.

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• Proof for the elements a and e in a loop G, with
  e the identity, there exist elements x and y s.t.
  ax e and za e. (divisibility, identity).
  Then ax za. Hit the right side of each with x,
  so (ax)x (za)x. But we have associativity now,
  so (ax)x z(ax). But ax e, so we have x z.
  Therefore the left and right inverses of a are
  the same.

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Constructing the Group
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Moving on
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Symmetric Magmi
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Symmetric Magmi

• A magma is symmetric if it is both right
  symmetric and left symmetric, i.e., if
• a(bc) (ac)b (right symmetric law)
• a(bc) (ba)c (left symmetric law)

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Symmetric Magmi

• A magma is symmetric if it is both right
  symmetric and left symmetric, i.e., if
• a(bc) (ac)b (right symmetric law)
• a(bc) (ba)c (left symmetric law)
• If it satisfies (1) or (2) but not both, it is
  called right or left symmetric

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• You can learn things about rings by looking at
  the multiplicative magma of the ring

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Theorem 2

• Every right (left) symmetric commutative magma is
  a commutative semigroup.

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Theorem 2

• Every right (left) symmetric commutative magma is
  a commutative semigroup.
• Proof

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Theorem 2

• Every right (left) symmetric commutative magma is
  a commutative semigroup.
• Proof
• a(bc) a(cb) by commutativity
• a(cb) (ab)c by the right symmetric law
• Therefore we have associativity, which is all we
  need for a magma to be a semigroup.

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• Theorem 20 Every symmetric ring R, in which xx
  0 implies x 0, is an associative-commutative
  ring

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• Theorem 20 Every symmetric ring R, in which xx
  0 implies x 0, is an associative-commutative
  ring
• (Note adjectives, such as symmetric, apply to a
  ring R if they apply to the multiplicative magma
  of R)

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Proof
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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)

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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)
• (ab)(ab) (b)(a)(ab) (a)(b)(ba)
  (ba)(ba)

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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)
• (ab)(ab) (b)(a)(ab) (a)(b)(ba)
  (ba)(ba)
• (ab)(ab) (a)(b)(ab) (b)(a)(ba)
  (ba)(ba) by (2)

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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)
• (ab)(ab) (b)(a)(ab) (a)(b)(ba)
  (ba)(ba)
• (ab)(ab) (a)(b)(ab) (b)(a)(ba)
  (ba)(ba) by (2)
• (ab)(ab) (a)(ab)(b) (b)(ba)(a)
  (ba)(ba) by (2)
• (not commutativity)

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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)
• (ab)(ab) (b)(a)(ab) (a)(b)(ba)
  (ba)(ba)
• (ab)(ab) (a)(b)(ab) (b)(a)(ba)
  (ba)(ba) by (2)
• (ab)(ab) (a)(ab)(b) (b)(ba)(a)
  (ba)(ba) by (2)
• (not commutativity)
• (ab)(ab) (ab)(ab) (ba)(ba) (ba)(ba) by
  (1)

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• (ab ba) 2 (ab)(ab) (ba)(ab) (ab)(ba)
  (ba)(ba)
• (ab)(ab) (b)(a)(ab) (a)(b)(ba)
  (ba)(ba)
• (ab)(ab) (a)(b)(ab) (b)(a)(ba)
  (ba)(ba) by (2)
• (ab)(ab) (a)(ab)(b) (b)(ba)(a)
  (ba)(ba) by (2)
• (not commutativity)
• (ab)(ab) (ab)(ab) (ba)(ba) (ba)(ba) by
  (1)
• 0

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• So, we have (ab ba) 2 0
• If this fact implies ab ba 0, then ab ba,
• So R is commutative.
• Theorem 2 gives us associativity as well.

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Remember the Ideal?

• Def an ideal I is a submagma of a magma G such
  that for all x in I and all g in G, xg and gx
  are in I.

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Theorem 7

• The center of a right (left) symmetric magma G is
  an ideal of G.
• Proof
• Let C be the center of G.
• If a,b are in G and c is in C, then (1) implies
• (cb)a c(ab) (1) (ab)c a(cb) (1)
• Thus (cb) is in C and (bc) (cb) is in C.
• Hence C is an ideal of G

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Theorem 21

• The center of a right (left) symmetric ring R is
  just the center of the multiplicative magma of R
  and is and ideal of R.

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• If cx xc for all x in R, then (1) implies
• (ab)c a(cb) a(bc), since c commutes with all
  elements in R. And,
• c(ab) (cb)a (1) (bc)a b(ac) (1) b(ca)
• (ba)c (1) c(ba) (ca)b (1)
• Recall that the center C is a subring of R.
• Thus from theorem 7 and the above fact, we have
  our desired result.

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HOMEWORK

• Prove that a quasigroup is a magma whose
  multiplication table is a latin square
• Find a loop that aint a group.
• The paper
• http//www.jstor.org/view/0025570x/di021070/02p018
  0p/0?currentResult0025570x2bdi0210702b02p0180p
  2b02cFFsearchUrlhttp3A2F2Fwww.jstor.org2Fse
  arch2FBasicResults3Fhp3D2526si3D126gw3Djtx
  26jtxsi3D126jcpsi3D126artsi3D126Query3Dsymm
  etric2Brings2Bgroupoids26wc3Don