The Fermi and Bose congruences for free semigroups on two generators
Allen Parks · Journal of Mathematical Physics · 1992
Using a free semigroup and a free monoid generated by two symbols, two congruences are introduced. These congruences provide natural equivalence classifications of all finite composite creation and annihilation operators for fermions and bosons such that equivalent operators act identically (to within a nonnegative integer multiple) upon the associated basis states. The resulting quotient structures are shown to be isomorphic to two elementary inverse semigroups and their algebras are used to construct generalized commutation forms. These provide for the easy evaluation of any fermion anticommutator and any reduced word boson commutator.