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Algebra II
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Monoids and groups Morphisms
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 1 / 8
Interactive
Algebra
Algebra 2
Monoids and groups
Morphisms
A.M. Cohen, H. Cuypers, H. Sterk
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
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
Interactive
Algebra