Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free (d-union-free, for short) languages. In this paper λ-free nondeterministic variants of 1CFPAs are used to define n-union-free languages. The defined language class is shown to be properly between the classes of (nondeterministic) union-free and d-union-free languages (in case of at least binary alphabet). In case of unary alphabet the class of n-union-free languages coincides with the class of union-free languages. Some properties of the new subregular class of languages are discussed, e.g., closure properties. On the other hand, a regular expression is in union normal form if it is a finite union of union-free expressions. It is well known that every regular expression can be written in union normal form, i.e., all regular languages can be described as finite unions of (nondeterministic) union-free languages. It is also known that the same fact does not hold for deterministic union-free languages, that is, there are regular languages that cannot be written as finite unions of d-union-free languages. As an important result here we show that every regular language can be defined by a finite union of n-union-free languages. This fact also allows to define n-union-complexity of regular languages.

中文翻译：

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重新审视无联合——确定性和非确定性无联合语言之间

无联合表达式是不使用联合运算的正则表达式。因此，（非确定性）无联合语言由仅使用串联和 Kleene 星号的正则表达式来描述。语言类的特征还在于一类特殊的有限自动机：1CFPAs 从它们的每个状态都有一个无循环的接受路径。显然，这样的自动机只有一个接受状态。此类自动机的确定性对应物定义了确定性无联合（d-union-free，简称 d-union-free）语言。在本文中λ- 1CFPAs 的非确定性变体用于定义n-无联合语言。已定义的语言类显示在（非确定性）无联合语言和无d-联合语言的类之间（至少在二进制字母表的情况下）。对于一元字母表，n-union-free 语言类与 union-free 语言类一致。讨论了新的子常规类语言的一些属性，例如闭包属性。另一方面，如果正则表达式是无联合表达式的有限联合，则它是联合范式。众所周知，每个正则表达式都可以写成联合范式，即所有正则语言都可以描述为（非确定性）无联合语言的有限联合。众所周知，相同的事实不适用于确定性无联合语言，即 有些常规语言不能写成无 d 联合语言的有限联合。作为这里的一个重要结果，我们展示了每种常规语言都可以由 n-union-free 语言的有限联合来定义。这一事实也允许定义正则语言的 n-union-complexity。