LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Definition 1. Define the relation ∼⊆V ∗ × V ∗ by
On Languages of Cyclic Words 279
x ∼ y ⇔ x = αβ ∧ y = βα for some α, β ∈ V ∗
Lemma 1. The relation ∼ is an equivalence.
Proof. Trivially, ∼ is reflexive since x = λx = xλ = x, and symmetric by
definition.
∼ is also transitive since x ∼ y, y ∼ z implies x = αβ, y = βα = γδ, z = δγ.
If |β| ≤|γ| then γ = βρ, α = ρδ, and therefore x = ρδβ, z = δβρ. Hence
x ∼ z
If |β| > |γ| then β = γσ, δ = σα, and therefore x = αγσ, z = σαγ. Hence
z ∼ x.
Thus ∼ is an equivalence relation. ✷
Definition 2. Denote by [x] the equivalence class of x consisting of all cyclic
permutations of x, andbyCV = V ∗ / ∼ the set of all equivalence classes of ∼.
Definition 3. For each cyclic word [x] a norm may be defined in a natural way
by ||[x]|| = |x|. Clearly, ||[x]|| is well defined since |ξ| = |x| for all ξ ∈ [x]. The
norm may be extended to sets of cyclic words by
||A|| = max{||[x]|| | [x] ∈ A}.
It is obvious from the definition that ||{[x]}◦{[y]}|| = ||{[x]}|| + ||{[y]}||, and
therefore ||A ◦ B|| ≤ ||A|| + ||B||.
The next aim is to define an associative operation on 2 CV .
Definition 4. Define an operation ⊙ on 2 CV as follows:
{[x]}⊙{[y]} = {[ξη] | ξ ∈ [x],η ∈ [y]}.
Note that {[λ]}⊙{[x]} = {[x]}⊙{[λ]} = {[x]}.
Unfortunately, we see that ⊙ is only commutative but not associative.
Lemma 2. The operation ⊙ is a commutative but not an associative.
Proof. {[x]}⊙{[y]} = {[ξη] | ξ ∈ [x],η ∈ [y]}
= {[ηξ] | ξ ∈ [x],η ∈ [y]} = {[y]}⊙{[x]}.
Thus ⊙ is commutative.
({[ab]}⊙{[c]}) ⊙{[d]} = {[abc], [acb]}⊙{[d]}
= {[abcd], [adbc], [abdc], [acbd], [adcb], [acdb]},
{[ab]}⊙({[c]}⊙{[d]}) ={[ab]}⊙{[cd]} = {[abcd], [acdb], [abdc], [adcb]}.
Thus ⊙ is not associative. ✷
Another operation ⊗ can be defined as follows.