Uniqueness of knot roots in $F\times I$ and prime decompositions of virtual knots

We introduce four types of reduction on the set of knots in thickened surfaces, i.e., in three-dimensional manifolds of the form $F\times I$, where $F$ is a closed orientable surface and $I=[0,1]$. It is proved that the process of applying these reductions to an arbitrary knot in a thickened surface is always finite. The resulting set of knots in thickened surfaces depends on the initial knot only up to the removal of trivial knots in thickened spheres. Reductions of knots in thickened surfaces induce the operation of connected summation of virtual knots. It is proved that every virtual knot can be decomposed into a connected sum of several prime or trivial virtual knots and the prime summands of the decomposition are defined uniquely.