The Finite Basis Problem For The Variety Generated By All Involution Semigroups Of Order N

Let（S_n,^*）and（M_n,^*）be the varieties generated by all involution semigroups and all involution monoids of order n respectively.In this paper,it is shown that both（S_n,^*）and（M_n,^*）are finitely based if and only if n≤4 respectively,then the finite basis prob-lems for the varieties（S_n,^*）and（M_n,^*）are completely solved.Furthermore,this paper investigates the counting problems of subvarieties of（S_n,^*）and（M_n,^*）and proves that（M_n,^*）has continuum many subvarieties if and only if n≥5;（S₂,^*）and（S₃,^*）have countably many subvarieties,（S_n,^*）has continuum many subvarieties when n≥5.The counting problem of subvarieties of（S₄,^*）remains to be solved.