Union-free languages are described by regular expressions using only concatenation and Kleene-star. Every regular language can be given as a union of finitely many union-free languages. By the minimal number of union-free languages needed in such union-free decompositions of a regular language, its union-complexity is defined. In this paper, the union-complexity of the languages obtained by various operations is studied, e.g., having two languages with union-complexities n and m, respectively, what could be the union-complexity of their union/concatenation/shuffle. Particularly, it is shown that the Kleene-star of any regular language has union-complexity 1. In some cases, e.g., at union and concatenation, the resulting language has a bounded union-complexity. In some other cases, e.g., at complement, the resulted language can have arbitrarily large union-complexity. At (k-th) power of a language, the case of the unary alphabet and the general case (alphabet with at least two symbols) have different upper bounds. While, in case of shuffle, there is an unbounded growth in the general case, while for languages over the unary alphabet the growth is bounded. Tight upper bounds are shown for all of the above mentioned cases (whenever the growth is bounded). At intersection we also show an unbounded growth in the general case, especially, over a binary alphabet.

无联合语言由仅使用连接和 Kleene-star 的正则表达式描述。每种常规语言都可以作为有限多个无联合语言的联合给出。通过在常规语言的这种无联合分解中所需的最小数量的无联合语言，它的联合复杂度被定义。本文研究了通过各种操作获得的语言的联合复杂度，例如，有两种语言的联合复杂度为 n和m，分别是什么可能是他们的联合/连接/洗牌的联合复杂性。特别是，它表明任何常规语言的 Kleene-star 具有联合复杂度 1。在某些情况下，例如在联合和连接时，生成的语言具有有限联合复杂度。在其他一些情况下，例如在补码时，生成的语言可以具有任意大的联合复杂度。在 ( k-th) 语言的力量，一元字母表的大小写和一般大小写（至少有两个符号的字母表）具有不同的上限。而在洗牌的情况下，一般情况下会有无限的增长，而对于一元字母表上的语言，增长是有界的。上面提到的所有情况都显示了严格的上限（只要增长是有界的）。在交集处，我们还展示了一般情况下的无限增长，尤其是在二进制字母表上。