Let L denote the Q-lattice of the variety \({\mathcal {V}}\) of lattices, i.e. the lattice of quasivarieties that are contained in \({\mathcal {V}}\). Let F denote the free lattice in \({\mathcal {V}}\) with \(\omega \) free generators and let Q(F) denote the quasivariety of lattices generated by F. Let Fin denote the collection of finite lattices which belong to Q(F) and let Q(Fin) denote the quasivariety generated by Fin. Moreover, let \({\mathcal {M}}^{-}_{3}\) denote the quasivariety of lattices which do not contain an isomorphic copy of \(M_3\) (the 5-element non-distributive modular lattice) as a sublattice and let \({\mathcal {S}}\) denote a selector of non-isomorphic finite quasicritical lattices which belong to Q(Fin). In this paper, we establish the following:

• The filter in L generated by Q(F) is prime.

• For every quasivariety \({\mathcal {K}}\) contained in \({\mathcal {M}}^{-}_{3}\), the interval \([{\mathcal {K}}, {\mathcal {V}}]\) contains an isomorphic copy of the ideal lattice I(F) of F. In particular, the filter in L generated by Q(F) contains an isomorphic copy of I(F).

• The distributive lattice of the order ideals of \(({\mathcal {S}}, \le )\), where \(A \le B\) means \(A \in Q(B)\), is uncountable and is a homomorphic image of the Q-lattice of Q(Fin).

让大号分别表示的各种Q-晶格\（{\ mathcal {V}} \）晶格的，即包含在quasivarieties的晶格\（{\ mathcal {V}} \） 。令F表示具有\（\ omega \）自由生成器的\（{\ mathcal {V}} \}中的自由晶格，而令Q（F）表示由F生成的晶格的准性。设Fin表示属于Q（F）的有限晶格的集合，设Q（Fin）表示Fin产生的准性。而且，让\（{\ mathcal {M}} ^ {-} _ {3} \）表示不包含\（M_3 \）（5元素非分布模块化晶格）的同构副本的晶格的准度。sublattice和let \ {{\ mathcal {S}} \\}表示属于Q（Fin）的非同构有限拟临界晶格的选择器。在本文中，我们建立了以下内容：

• 由Q（F）生成的L中的滤波器是素数。

• 对于\（{\ mathcal {M}} ^ {-} _ {3} \）中包含的每一个准\\ {{mathcal {K}} \ }，区间\（[{{mathcal {K}}，{ \ mathcal {V}}] \）包含理想的晶格同构复制我（˚F的）˚F。特别是，由Q（F）生成的L中的滤波器包含I（F）的同构副本。

• \（（{{mathcal {S}}，\ le）\）的阶理想分布矩阵，其中\（A \ le B \）表示\（A Q（B）\）是不可数的，是Q（Fin）的Q晶格的同态图像。