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What are the basic properties of products in TOC?
It is easy to see that for any language L the following simple properties hold −
L · {∧} = {∧} · L = L
L · ∅ = ∅ · L = ∅
Now let’s see the commutativity and associativity of the operation of concatenation.
Properties of products – commutativity
The operation of concatenation is not commutative. In other words, the order matters!
Given two languages L and M, it’s usually true that
L · M ≠ M · L
Example
If L = {ab, ac} and M = {a, bc, abc}, then the product
L · M is the language
L · M = {aba, abbc, ababc, aca, acbc, acabc},
But the product M · L is the language
M · L = {aab, aac, bcab, bcac, abcab, abcac}
These have no strings in common!
Properties of products – associativity
The operation of concatenation is associative. In other words, if L, M, and N are languages, then
L · (M · N) = (L · M) · N
Example
If L = {a, b}, M = {a, aa} and N = {c, cd}, then
L · (M · N) = L · {ac, acd, aac, aacd}
= {aac, aacd, aaac, aaacd, bac, bacd, baac, baacd}.
This is the same as,
(L · M) · N = {aa, aaa, ba, baa} · N
= {aac, aacd, aaac, aaacd, bac, bacd, baac, baacd}.
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