Barycentrically associative and preassociative functions

Abstract

ABSTRACT. We investigate the barycentric associativity property for functions with indefinite arities and discuss the more general property of barycentric preassociativity, a generalization of barycentric associativity which does not involve any composition of functions. We also provide a generalization of Kolmogoroff-Nagumo's characterization of the quasi-arithmetic mean functions to barycentrically preassociative functions.

Strongly barycentrically associative and preassociative functions

“…As already observed in [9], if a B-associative operation F ∶ X * → X ∪ {ε} is such that ran(F n ) ⊆ X for every n ⩾ 1, then the value of F (ε) is unimportant in the sense that if we modify this value, then the resulting operation is still B-associative. Clearly, this observation also holds for strongly B-associative operations, B-preassociative functions, and strongly B-preassociative functions.…”

“…Actually, it can be shown [9] that if an operation F ∶ X * → X ∪ {ε} is B-associative then it is both B-preassociative and arity-wise range-idempotent. The converse result holds whenever ran(F n ) ⊆ X for every n ⩾ 1 (note that this latter condition was wrongly omitted in [9]). The following proposition shows that this result still holds if we replace B-associativity and B-preassociativity by their strong versions.…”

“…It is easy to see that any B-associative operation F ∶ X * → X ∪ {ε} is necessarily B-preassociative [9]. This observation motivates the introduction of the following property, which generalizes strong B-associativity.…”

“…Finally, a variadic function F ∶ X n → Y is said to be ε-standard [8] if ε ∈ Y and F (x) = ε ⇔ x = ε. Recall that a variadic operation F ∶ X * → X ∪ {ε} is said to be barycentrically associative (or B-associative for short) [9] if it satisfies the equation…”

“…Recall that a variadic function F ∶ X * → Y is said to be barycentrically preassociative (or B-preassociative for short) [9] if, for every xyy ′ z ∈ X * , we have…”