Regularity Conditions

Abstract

A language L over the finite alphabet A is recognizable or regular if and only if the syntactic monoid S(L) is finite (see Theorem 4.1.1). A property P of languages is said to be syntactic if whenever two languages L and L′ have the same syntactic monoid and Lhas the property P, then also L′ has the property P. In this sense, commutativity is a syntactic property (L is commutative if and only if its syntactic monoid is commutative), periodicity is syntactic and also rationality is a syntactic property. We point out that when a characterization of rationality involves only properties of languages which are syntactic, then this result is more a result on semigroups than on languages. Indeed, a relation ${{f}_{1}}{{ \equiv }_{L}}{{f}_{2}}$, with f₁, f₂ ∈ A^*, can be rewritten as

$$\forall u,\upsilon \in A*(u{{f}_{1}}\upsilon \in L \Leftrightarrow u{{f}_{2}}\upsilon \in L).$$

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