Abstract
This chapter illustrates the ideas inspired by Hilbert's Sixth Problem and describes new directions followed today in research on quantum structures. In specific, Hilbert's Sixth Problem lies between mathematics and physics. The Sixth Problem involves finding a few physical axioms, similar to the axioms of geometry, and that can describe a theory for a class of physical events that is as large as possible. This chapter presents the elements of certain new algebraic quantum structures for which commutativity of addition is not assumed. It focuses mainly on pseudo effect algebras and on pseudo MV-algebras. It also explores how the algebraic structure of a pseudo effect algebra can determine the whole unital po-group. The chapter concludes that in view of the particular nature of quantum measurements, the relationship between quantum mechanics and the mathematics of quantum structures is similar to that between statistical physics and classical probability theory. The investigation of quantum structures can bring about new and useful results for both mathematics and quantum mechanics as well as for their applications.
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