The ideal induction of vector fields in fluids and plasmas is presented as invariance under local convection, 'freezing' into flowlines and transfer of potential. Related consequences are discussed, namely the conservation of local flux and total helicity and the force-free relaxation state. In the framework of single-fluid Hall-MHD plasma flow model, the magnetic field and vorticity (which are formally analogous and generally anti-correlated, but each not ideally inducted, i.e. perfectly 'frozen-into' the flow) are shown to combine in a unified magneto-vorticity field, which is ideally inducted in perfectly-conducting, however even forced, non-isentropic and viscous plasmas. Relaxation plasma states of conserved or extreme helicity magnetovorticity fields are derived and shown to be generalized force-free states, similar to those previously derived in the framework of Hall-MHD and the multicomponent plasma model. The magneto-vorticity induction in visco-resistive plasmas is also discussed. Application of the magneto-vorticity field concept in the study of type I superconductors and the spontaneous generation of magnetic fields are reviewed. The Cowling 'anti-dynamo' theorem for axisymmetric flows is extended in Hall-MHD and for arbitrary flows and is shown that, in principle, the resistive (ohmic) dissipation of the magnetic field can be balanced by non-isentropic heating and/or helical forcing effects.