Cônes rationnels commutatifs

Every commutative quasirational language is regular

“…Using the results of Eerstel and Boasson ( [2], [4]) Latteux proves in [9] that the theorem is true when k = 2.…”

“…The rank of T, denoted by rank (7), is s if T=S X U • • • U S m where each S t is a linear set and max rank (S f ) = s. It can i be verified that the rank of each semilinear set is uniquely determined. The convex closure conv (S) of the linear set S is defined by: u 1 + ... +<x r u r | OLJSQ, OLJ^0J=1 9 .. Ginsburg proves in [6] that: (i) the intersection of two semilinear sets is a semilinear set; (ii) the complement of a semilinear set is a semilinear set; and (iii) each semilinear set is a finite union of proper linear sets. These facts are extensively used in our proofs.…”

“…4-... + oc"u n for some a,e6, i = 1, ..., n. Since conv^J nconv(T 1 ) = 0, there is at least oneje{l, ..., n} such that a 7 < 0. Let: a = max{|a J .| | OLJ<0J=1 9 ..., n}.…”

“…Let ,T(JSf) (i"(JSf)) dénote the (full) trio generated by the language family jSf. In [1], [9] and [10] we can find the following conjecture: CONJECTURE 1: If L is a nonregular language in c(ât) 9 then Df is in #*(L). We show that Df is in 2T(L) for each nonregular language L in c{M) thus proving the conjecture.…”

“…The minimality of languages is studied in several articles, for instance in [1], [3], [9] and [10]. Let ,T(JSf) (i"(JSf)) dénote the (full) trio generated by the language family jSf.…”