This paper is part of the investigation of some operations on words and languages with motivations coming from DNA biochemistry, namely three variants of hairpin completion and three variants of hairpin reduction. Since not all the hairpin completions or reductions of semilinear languages remain semilinear, we study sufficient conditions for semilinear languages to preserve their semilinearity property after applying the non-iterated hairpin completion or hairpin reduction. A similar approach is then applied to the iterated variants of these operations. Along these lines, we define the hairpin reduction root of a language and show that the hairpin reduction root of a semilinear language is not necessarily semilinear except the universal language. A few open problems are finally discussed.

本文是对某些单词和语言操作进行研究的一部分，其动机来自DNA生物化学，即发夹完成的三个变体和发夹减少的三个变体。由于并非所有的半线性语言的发夹完成或归约都保持半线性，因此我们研究了半线性语言的充分条件，可以在应用非迭代的发夹完成或减少发夹后保持其半线性属性。然后将类似的方法应用于这些操作的迭代变体。沿着这些思路，我们定义了一种语言的发夹减少根，并表明半线性语言的发夹减少根除了通用语言外不一定是半线性的。最后讨论了一些未解决的问题。