Algebraic Combinations of Flows II

Abstract

There is an important difference between the theory of maxima and minima of members of a family Rep ⁺_q (ϕ), and the theory of harmonic sums and sums. In dealing with the maximum or minimum of two members ψ;, π of a family Rep ⁺_q (ϕ), we had only to ask when ψ ⋁ π and ψ ⋀ π could exist and how they behaved. The meaning of the symbols ψ ⋁ π and ψ ⋀ π was preassigned by the partial order ≤ in Rep ⁺_q (ϕ). However, the harmonic sum and sum need to be defined in such a way that they have the behavior we expect of them. In the case of harmonic sums, the definition given in Section 8.35 was “justified” by the equation

$$ (u\, \oplus \,v)\,\varphi \, = \,u\,\varphi \oplus \,v\,\varphi . $$