The ideal induction of vector fields in fluids and plasmas is presented asinvariance under local convection, `freezing' into flowlines and transfer ofpotential. Related consequences are discussed, namely the conservation oflocal flux and total helicity and the force-free relaxation state. In theframework of single-fluid Hall-MHD plasma flow model, the magnetic field andvorticity (which are formally analogous and generally anti-correlated, buteach not ideally inducted, i.e. perfectly `frozen-into' the flow) are shown tocombine in a unified magneto-vorticity field, which is ideallyinducted in perfectly-conducting, however even forced, non-isentropic and viscousplasmas. Relaxation plasma states of conserved or extremehelicity magneto-vorticity fields are derived and shown to be generalizedforce-free states, similar to those previously derived in the framework ofHall-MHD and the multi-component plasma model. The magneto-vorticity inductionin visco-resistive plasmas is also discussed. Application of themagneto-vorticity field concept in the study of type I superconductors and thespontaneous generation of magnetic fields are reviewed. The Cowling`anti-dynamo' theorem for axisymmetric flows is extended in Hall-MHD and forarbitrary flows and is shown that, in principle, the resistive (ohmic)dissipation of the magnetic field can be balanced by non-isentropic heatingand/or helical forcing effects.