A recently introduced operation of geometrical closure on formal languages is investigated from the viewpoint of algebraic language theory. Positive varieties $\mathcal{V}$ containing exclusively languages with regular geometrical closure are fully characterised by inclusion of $\mathcal{V}$ in $\mathcal{W}$, a known positive variety arising in the study of the commutative closure. It is proved that the geometrical closure of a language from the intersection of $\mathcal{W}$ with the variety of all star-free languages $\mathcal{SF}$ always falls into ${\mathcal{R}}_{LT}$, which is introduced as a subvariety of $\mathcal{R}$, the variety of languages recognised by R-trivial monoids. All classes between ${\mathcal{R}}_{LT}$ and $\mathcal{W}\cap \mathcal{SF}$ are thus geometrically closed: for instance, the level 3/2 of the Straubing-Thérien hierarchy, the DA-recognisable languages, or the variety $\mathcal{R}$. It is also shown that $\mathcal{W}\cap \mathcal{SF}$ is the largest geometrically closed positive variety of star-free languages, while there is no largest geometrically closed positive variety of regular languages.

从代数语言理论的角度研究了最近在形式语言上引入的几何闭包运算。正品种$\mathcal{五}$仅包含具有规则几何闭合的语言的完全特征在于包含$\mathcal{五}$在$\mathcal{W}$，在交换闭包的研究中出现的一个已知的正变体。从交集证明了语言的几何闭合$\mathcal{W}$各种无星级语言$\mathcal{顺丰}$总是陷入${\mathcal{R}}_{\mathrm{大号}吨}$, 它作为一个子变体引入$\mathcal{R}$, 被R-平凡幺半群识别的各种语言。之间的所有班级${\mathcal{R}}_{\mathrm{大号}吨}$和$\mathcal{W}\cap \mathcal{顺丰}$因此在几何上是封闭的：例如，Straubing-Thérien 层次结构的第 3/2 级、DA可识别的语言或变体$\mathcal{R}$. 这也表明$\mathcal{W}\cap \mathcal{顺丰}$是无星语言的最大几何闭合正变体，而正则语言没有最大几何闭合正变体。