Question

State and prove generalized commutative law in a commutative semigroup

Answer 1

Generalized commutative law (GCL) :

let {i₁, i₂, ..., i_n} be the permutation of {1, 2, .., n}, then for elements a₁, a₂, ... a_n of the semigroup we have

$$a_1*a_2*...*a_n=a_{i_1}*a_{i_2}*...*a_{i_n}.$$

Proof (using extended mathematical induction):

1) For n = 1 we have a₁ = a₁, for n = 2 a₁*a₂ = a₂*a₁ therefor GCL holds for n = 1 and n = 2.

2)Assume that GCL is true for all n < k + 1.

3)Prove that GCL is true for n = k + 1, that is

$$a_1*a_2*...*a_{k+1}=a_{i_1}*a_{i_2}*...*a_{i_{k+1}}.$$

Let

$$a_{i_m}=a_{k+1}$$

then using generalized associative property of the semigroup

$$a_{i_1}*a_{i_2}*...*a_{i_{m-1}}*a_{i_{m}}*a_{i_{m+1}}*...*a_{i_{k+1}}=$$

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}})*(a_{i_{m}}*(a_{i_{m+1}}*...*a_{i_{k+1}}))$$

using 2)

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}})*(a_{i_{m}}*(a_{i_{m+1}}*...*a_{i_{k+1}}))=$$

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}})*((a_{i_{m+1}}*...*a_{i_{k+1}})*a_{i_{m}})$$

using generalized associative property again

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}})*((a_{i_{m+1}}*...*a_{i_{k+1}})*a_{i_{m}})=$$

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}}*a_{i_{m+1}}*...*a_{i_{k+1}})*a_{i_{m}}$$

(i₁, i₂,..., i_m-1, i_m+1,...,i_k+1 is a permutation of 1, 2 , ..., k) using 2) for n = k

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}}*a_{i_{m+1}}*...*a_{i_{k+1}})=$$

$$a_1*a_2*...*a_k$$

therefor

$$(a_{i_1}*a_{i_2}*...*a_{i_{m-1}}*a_{i_{m+1}}*...*a_{i_{k+1}})*a_{i_{m}}=$$

$$a_1*a_2*...*a_k*a_{i_m}=a_1*a_2*...*a_k*a_{k+1}$$

and

$$a_{i_1}*a_{i_2}*...*a_{i_{k+1}}=$$

$$a_{i_1}*a_{i_2}*...*a_{i_{m-1}}*a_{i_{m}}*a_{i_{m+1}}*...*a_{i_{k+1}}=$$

$$a_1*a_2*...*a_k*a_{k+1}$$

So we have proved GCL using extended mathematical induction.