Algebra II - Monoids and Groups

Monoids and groups Morphisms

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 1 / 8

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Algebra

Algebra 2

Monoids and groups

Morphisms

A.M. Cohen, H. Cuypers, H. Sterk

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Algebra

Monoids and groups Morphisms

The standard notion for comparing structures isthat of homomorphism.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 2 / 8

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Algebra

Monoids and groups Morphisms

Definition HomomorphismLet S1 and S2 be two structures with ni -ary op-erations ∗i,1 and ∗i,2, respectively (where i runsthrough a finite set).A homomorphism between these structures is amap f : S1→S2 respecting all operations, i.e., forall i we have

f (∗i,1(a1, . . ., ani ))=∗i,2(f (a1), . . ., f (ani )).

If f is bijective,then we call f an isomorphism.In particular, for monoids [M1, · 1, e1] and[M2, · 2, e2] this means the following.A homomorphism between M1 and M2 is a mapf : M1→M2 with the following properties.

f (e1)=e2

for all a, b: f (a · 1b)=f (a) · 2f (b)

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 8

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Algebra

Monoids and groups Morphisms

Definition HomomorphismLet S1 and S2 be two structures with ni -ary op-erations ∗i,1 and ∗i,2, respectively (where i runsthrough a finite set).A homomorphism between these structures is amap f : S1→S2 respecting all operations, i.e., forall i we have

f (∗i,1(a1, . . ., ani ))=∗i,2(f (a1), . . ., f (ani )).

If f is bijective,then we call f an isomorphism.In particular, for monoids [M1, · 1, e1] and[M2, · 2, e2] this means the following.A homomorphism between M1 and M2 is a mapf : M1→M2 with the following properties.

f (e1)=e2

for all a, b: f (a · 1b)=f (a) · 2f (b)

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 8

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Algebra

Monoids and groups Morphisms

Definition HomomorphismLet S1 and S2 be two structures with ni -ary op-erations ∗i,1 and ∗i,2, respectively (where i runsthrough a finite set).A homomorphism between these structures is amap f : S1→S2 respecting all operations, i.e., forall i we have

f (∗i,1(a1, . . ., ani ))=∗i,2(f (a1), . . ., f (ani )).

If f is bijective,then we call f an isomorphism.In particular, for monoids [M1, · 1, e1] and[M2, · 2, e2] this means the following.A homomorphism between M1 and M2 is a mapf : M1→M2 with the following properties.

f (e1)=e2

for all a, b: f (a · 1b)=f (a) · 2f (b)

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 8

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Algebra

Monoids and groups Morphisms

If two structures are isomorphic (that is, thereis an isomorphism from one to the other), thenthey are of the ’same shape’ (morph = shape).An isomorphism S1→S1 (that is, with both do-main and target structure the same) is called anautomorphism of S1.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 8

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Algebra

Monoids and groups Morphisms

Example Suppose that all elements of the monoid M canbe expressed as products of a single element, sayc . So M={c0, c , c2, c3, . . .}. Then the monoidis said to be generated by c .Define a map f : N→M by f (n)=cn. Then wehave f (n + m) = cn+m = cn·cm = f (n)·f (m).Also, f (0)=1. Hence f is a homomorphism ofmonoids. Clearly, f is surjective. But it need notbe injective. If M is a free monoid, then the mapf is also injective. Another example of a homo-morphism of monoids is the length function for afree monoid. Indeed, if M is a free monoid overan alphabet A, then the length function L from Mto N satisfies L(∅)=0 and L(xˆy)=L(x) + L(y).If A has size 1, this length function is the inverseof the homomorphism f .

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 5 / 8

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Algebra

Monoids and groups Morphisms

Remark The notion of homomorphism of semi-groups issimilar; the condition about the identity elementis dropped, of course. Notions like homomor-phisms, isomorphisms, and automorphisms existfor all structures. We shall encounter them againwhen we discuss rings, groups, and fields.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 6 / 8

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Algebra

Monoids and groups Morphisms

Theorem If f : M1→M2 is an isomorphism of monoids, then

1 the cardinalities of M1 and M2 are equal;

2 the inverse map f −1 : M2→M1 is also anisomorphism of monoids.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 7 / 8

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Algebra

Monoids and groups Morphisms

Theorem If f : M1→M2 is an isomorphism of monoids, then

1 the cardinalities of M1 and M2 are equal;

2 the inverse map f −1 : M2→M1 is also anisomorphism of monoids.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 7 / 8

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Algebra

Monoids and groups Morphisms

Theorem If f : M1→M2 is an isomorphism of monoids, then

1 the cardinalities of M1 and M2 are equal;

2 the inverse map f −1 : M2→M1 is also anisomorphism of monoids.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 7 / 8

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Algebra

Monoids and groups Morphisms

Example Consider the monoids [Monoids]C1,1 and C0,2,given by the following multiplication tables.

· 1 a1 1 aa a 1

· 1 b1 1 bb b b

Both have size 2. But they are not isomorphic.For otherwise, there would be an isomorphism:f : C1,1→C0,2 with f (1)=1. Hence, as f is bi-jective, also f (a)=a. But then we would have1 = f (1) = f (a2) = f (a2) = b2 = b, a contra-diction.

A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 8 / 8

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Algebra