| Summary: | We propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Such an approach allows us to interpret some of the combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo mathvariant="script">ℓ</mo><mn>1</mn></msub><mrow><mo>(</mo><msub><mi mathvariant="double-struck">Z</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, induced by the supersymmetric polynomials, and the semi-ring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semi-ring structures into this set and discuss their possible physical interpretations.
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