Formal Language Theory: Language Concatenation

The relation G is a unary relation on the groundset pN Y q.

If u G v, then u can be derived from v in one step.

The notation u G v means that v can be derived from u in one step.

A sentential form of a grammar G is a word that can be derived from the start symbol of G in one step.

If R and S are binary relations, then R S is also a binary relation.

The relation G is the same as the relation G.

The concatenation operation is commutative for all alphabets.

The equation u(pvw)q = (upv)wq holds for any u, v, w ∈ V*.

The set V* along with the concatenation operation is a group with a unit element ε.

The i-th power of a word u is the (i-1) fold concatenation of u by itself.

The concatenation operation is associative, meaning that (uv)w = u(vw) for any u, v, w ∈ V*.

The unit element ε for the concatenation operation is an empty string.

The concatenation of two languages is a commutative operation.

The empty string ε is a unit element of language concatenation.

The power of a language L with exponent 0 is equal to L itself.

The operation of language concatenation forms a group.

The iterative closure of a language L is denoted as L^+.

The associative property holds for language concatenation.