We investigate collections of reversible gates closed under parallel and serial composition, with and without borrow and ancilla closure. In order to better understand the structure of these collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice, that is, collections that are not all gates, but the addition of a single new gate will allow us to construct all gates. From previous work, we know the maximal closed sets over a finite alphabet.

In this paper we extend this work to ancilla and borrow closure for reversible gates. Here we find several structural results, where we know that there are restricted possibilities for all maximal borrow and ancilla closed collections of gates. We determine several classes of maximal gates and determine their dual structures as weights as defined by Jeřábek. These maximal closed collections of gates correspond to minimal weight permutation co-clones.

Maximality has several applications, one of the interesting ones being that a gate in no maximal closed set must be universal. Thus an understanding of maximality is intimately tied to an understanding of universality.

我们研究了在并行和串行组合下关闭的可逆门的集合，有和没有借位和辅助关闭。为了更好地理解这些可逆门集合的结构，我们研究了闭集的格子和这个格子的最大成员，即不是所有门的集合，但添加一个新的门将允许我们建造所有的门。从以前的工作中，我们知道有限字母表上的最大闭集。

在本文中，我们将这项工作扩展到可逆门的辅助和借用闭包。在这里，我们发现了几个结构结果，我们知道所有最大借用和辅助关闭门集合的可能性是有限的。我们确定了几类最大门，并将它们的双重结构确定为 Jeřábek 定义的权重。这些最大的封闭门集合对应于最小的权重排列共克隆。

极大化有多种应用，其中一个有趣的应用是没有极大闭集的门必须是通用的。因此，对极大性的理解与对普遍性的理解密切相关。