H-bounded and semi-discrete languages

Since every hypercode is finite, one may ask for the significance of the property that a language L admits a uniform upper bound on the size of hypercodes included in L . Such a language L is called h-bounded . We prove that a rational language L is h -bounded iff it is thin iff it is semi-discrete , i.e., L contains at most k words of any given length for some fixed k ∈ ℕ. Moreover, a representation of these languages by regular expressions is established. Concerning the general case, some properties of the syntactic monoid Synt( L ) of an h -bounded (semi-discrete) language are derived. If L is not disjunctive, then Synt( L ) contains a zero element. Every subgroup of Synt( L ) is a finite cyclic group. The idempotents of Synt( L )\{0, 1~ form an antichain with respect to the usual partial order.