Possibilities to have different kinds of quantum mechanics as example of deformations of Wigner-Moyal-Groenawald star-product

Quantizer-dequantizer formalism (O.V.Manko, V.I.Manko, G.Marmo J Phys A 2002)to define the rules of product of functions satisfying the associativity condition [ e.g.in arythmetics (2x3)x5=2x(3x5)] is reviewed.Well known formulation of quantum mechanics as deformation of usual point-wise product of functions on phase space replacing the product by nonlocal and noncommutative but associative product of the functions -observables given by the integral kernel (Groenewald kernel) is discussed. Recently found method (A.Ibort, V.I.Manko, G.Marmo, A.Simoni, C.Stornaiolo, F.Ventriglia, Phys.Scripta 2016) to obtain new rules of the function products from the initial rule is used to construct new versions of quantum mechanics on the base of deforming standard Moyal star-product.The tomographic probability representation of usual quantum mechanics and its corresponding deformations are also discussed.