Let \({\mathscr {C}}\! om \) denote the variety of all commutative semigroups. For \(n\ge 1\) let \({{\mathscr {N}}\! il }_n\) (respectively \({\mathscr {N}}_n\)) denote the variety of all nil (respectively nilpotent) semigroups of index n, and let \({\mathscr {N}}\! il \) (respectively \({\mathscr {N}}\)) denote the generalized variety of all nil (respectively nilpotent) semigroups. For a class \({\mathscr {W}}\) of semigroups let \({\mathscr {L}}({\mathscr {W}})\) denote the lattice of all varieties contained in \({\mathscr {W}}\), and let \({\mathscr {G}}({\mathscr {W}})\) denote the lattice of all generalized varieties contained in \({\mathscr {W}}\). Almeida has proved that the map \(\alpha :{\mathscr {L}}({\mathscr {N}}\! il \cap {\mathscr {C}}\! om )\cup \{{\mathscr {N}}\! il \cap {\mathscr {C}}\! om \}\rightarrow {\mathscr {G}}({\mathscr {N}}\cap {\mathscr {C}}\! om )\) given by \({\mathscr {W}}\alpha = {\mathscr {W}}\cap {\mathscr {N}}\) is an isomorphism, and asked whether the extension \({\alpha }': {\mathscr {L}}({\mathscr {N}}\! il ) \cup \{{\mathscr {N}}\! il \}\rightarrow {\mathscr {G}}({\mathscr {N}})\) of this map is also an isomorphism. The author has previously shown that \({\alpha }'\) is not injective, using varieties of nil index 5, and noted that the corresponding map \({\alpha }_n : {\mathscr {L}}({{\mathscr {N}}\! il }_n) \rightarrow {\mathscr {G}}({\mathscr {N}} \cap {{\mathscr {N}}\! il }_n)\) given by \({\mathscr {W}} {\alpha }_n ={\mathscr {W}}\cap {\mathscr {N}}\) is therefore not injective for \(n\ge 5\). The question of smaller values of n was left open, and in this article it is shown that \({\alpha }_n\) is not injective for \(n\ge 2\), using Leech’s square-free words.

令\（{\ mathscr {C}} \！om \）表示所有可交换半群的多样性。对于\（n \ ge 1 \），让\（{{\ mathscr {N}} \！il _n \）（分别\（{\ mathscr {N}} _ n \））表示所有nil的种类（分别索引n的nilpotent个半群，令\（{\ mathscr {N}} \！il \）（分别为\ {{\ mathscr {N}} \））表示所有nil个（分别为nilpotent）半群的广义变种。对于半组的类\（{\ mathscr {W}} \\}，让\（{\ mathscr {L}}（{\ mathscr {W}}）\）表示\（{\ mathscr {W}} \），然后\（{\ mathscr {G}}（{\ mathscr {W}}）\）表示\（{\ mathscr {W}} \\}中包含的所有广义变体的格。Almeida已证明地图\（\ alpha：{\ mathscr {L}}（{\ mathscr {N}} \！il \ cap {\ mathscr {C}} \！om）\ cup \ {{\ mathscr { N}} \！il \ cap {\ mathscr {C}} \！om \} \ rightarrow {\ mathscr {G}}（{\ mathscr {N}} \ cap {\ mathscr {C}} \！om） \）由下式给出\（{\ mathscr【W}} \阿尔法= {\ mathscr【W}} \帽{\ mathscr {N}} \）是一个同构，并询问是否扩展\（{\阿尔法}” ：{\ mathscr {L}}（{\ mathscr {N}} \！il）\ cup \ {{\ mathscr {N}} \！il \} \ rightarrow {\ mathscr {G}}（{{\ mathscr {此图的N}}）\）也是同构的。作者先前已经证明\（{\ alpha}'\）是非内射的，使用nil索引的变种5，并注意到对应的映射\（{\ alpha __n：{\ mathscr {L}}（{{\ mathscr {N}} \！il __n）\ rightarrow { \ mathscr {G}}（{\ mathscr {N}} \ cap {{\ mathscr {N}} \！il _n）\）由\（{\ mathscr {W}} {\ alpha} _n = {因此\ mathscr {W}} \ cap {\ mathscr {N}} \）对于\（n \ ge 5 \）来说不是内射性的。较小的n值的问题尚待解决，在本文中，使用Leech的无平方词表明\（{\ alpha __n \）对于\ {n \ ge 2 \ }而言不是内射性。