Vector subspaces of finite fields and star operations on pseudo-valuation domains
Let $F=\mathbb{F}_{q}$ be the field with $q$ elements, where $q$ is a prime power and $L=\mathbb{F}_{q^{n}}$ be its extension of degree $n$. We denote by $\mathcal{O}(L,F)$ the set of $F$-vector subspaces of $L$ and by $ \mathcal{O}_{t}(L,F)$ the set of subspaces having dimension $t$. The multiplicative group $L^{\ast}$ of $L$ acts on $\mathcal{O}_{t}(L,F)$. Let $ l(q,n,t)$ be the number of orbits of $\mathcal{O}_{t}(L,F)$ under this action. In this paper, the author provides an explicit fomula and some bounds for $l(q,n,t)$. These results are then applied to the study of the set of $F$ star operations on $L,$ $\mathrm{Star}_{F}(L)$. Recall that for $F\subset L $ an extension of fields, we say that a map $\ast: \mathcal{O} (L,F) \rightarrow \mathcal{O} (L,F); V \longmapsto V^{\ast}$, is a F-star operation on $L$ if, for every $V,$ $W\in \mathcal{O} (L,F)$ and every $\beta \in L,$ we have: \begin{itemize} \item[$\bullet$] $V\subseteq V^{\ast}$. \item[$\bullet$] if $V\subseteq W,$ then $V^{\ast}\subseteq W^{\ast}$. \item[$\bullet$] $(V^{\ast})^{\ast}=V^{\ast}$. \item[$\bullet$] $(\beta V)^{\ast}=\beta V^{\ast}$. \item[$\bullet$] $F=F^{\ast}$. \end{itemize} The author proves that if $R$ is a $PVD$ with associated valuation ring $V$ such that $R\neq V$ and if $F$, $L$ are the respective residue fields, then there is an order preserving bijection between the set of star operations on $R$ and the set of $F$-star operations on $L$. This result is in fact implicitly present in [\textit{M. H. Park}, Rocky Mt. J. Math. 42, No. 6, 1939--1951 (2012; Zbl 1264.13006)]. \par The author investigates the asymptotic behavior of $\left| \mathrm{Star}_{F}(L)\right| $ and gives the exact values of $\left| \mathrm{Star}_{F}(L)\right| $ in case $[L:F]=4$ or $[L:F]=5$.
scientific article; zbMATH DE number 7043368