Puoliinteger

Puoliinteger is a term used to denote numbers that are integer plus one-half, commonly known in English as half-integers. Formally, a puoliinteger has the form n + 1/2 where n belongs to the integers Z; examples include 1/2, −3/2 and 5/2. The set of puoliintegers can be written as Z + 1/2 and is a coset of the integer lattice under addition.

Algebraically, puoliintegers are not closed under addition or multiplication: the sum of two puoliintegers is an integer, while the product of two puoliintegers need not be a puoliinteger. Multiplying a puoliinteger by 2 yields an odd integer, and adding an integer to a puoliinteger returns another puoliinteger. These elements therefore behave as a shift of the integer group by 1/2 and are useful in modular and parity considerations.

Puoliintegers appear in several areas of mathematics and physics. In representation theory and quantum mechanics they describe half-integer spin values (e.g., spin-1/2 particles). In discrete mathematics and number theory they arise when working with lattice points offset by half a unit, and in Fourier analysis and signal processing they can index half-step shifts. The concept is primarily notational and structural, emphasizing the role of a constant offset from the integer set.

The term is often interchangeable with half-integer; usage and notation vary by language and context, but the underlying mathematical properties remain the same.