A numerical semigroup is a submonoid of \({{\mathbb {Z}}}_{\ge 0}\) whose complement in \({{\mathbb {Z}}}_{\ge 0}\) is finite. For any set of positive integers a, b, c, the numerical semigroup S(a, b, c) formed by the set of solutions of the inequality \(ax \bmod {b} \le cx\) is said to be proportionally modular. For any interval \([\alpha ,\beta ]\), \(S\big ([\alpha ,\beta ]\big )\) is the submonoid of \({{\mathbb {Z}}}_{\ge 0}\) obtained by intersecting the submonoid of \({{\mathbb {Q}}}_{\ge 0}\) generated by \([\alpha ,\beta ]\) with \({{\mathbb {Z}}}_{\ge 0}\). For the numerical semigroup S generated by a given arithmetic progression, we characterize a, b, c and \(\alpha ,\beta \) such that both S(a, b, c) and \(S\big ([\alpha ,\beta ]\big )\) equal S.

数值半群是一个子幺\（{{\ mathbb {Z}}} _ {\ GE 0} \），其补体在\（{{\ mathbb {Z}}} _ {\ GE 0} \）是有限. 对于任何一组正整数a ,  b ,  c，由不等式\(ax \bmod {b} \le cx\)的解集形成的数值半群S ( a ,  b ,  c )被称为成比例模块化。对于任何区间\([\alpha ,\beta ]\)，\(S\big ([\alpha ,\beta ]\big )\)是\({{\mathbb {Z}}}_{ \ge 0}\)通过与\({{\mathbb {Q}}}_{\ge 0}\)由\([\alpha ,\beta ]\)与\({{\mathbb {Z}}}_{\ge 0} \)。对于由给定等差数列生成的数值半群S，我们刻画a ,  b ,  c和\(\alpha ,\beta \)使得S ( a ,  b ,  c ) 和\(S\big ([\alpha ,\beta ]\big )\)等于S。