Pointed quandles of linkoids

In this paper we define the fundamental quandle of knotoids and linkoids and prove that it is invariant under the under forbidden-move and hence encodes only the information of the underclosure of the knotoid. We then introduce $n$-pointed quandles, which generalize quandles by specifying $n$ elements as ordered basepoints. This leads to the notion of fundamental pointed quandles of linkoids, which enhances the fundamental quandle. Using $2$-pointed quandle allows us to distinguish 1-linkoids with equivalent under-closures and leads to a couple of 1-linkoid invariants. We also generalize the notion of homogeneity of quandles to $n$-homogeneity of quandles. We classify all $\infty$-homogeneous, finite quandles.