Semigroup-valued multiplicative functions

Multiplicative complexity of vector valued Boolean functions

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called ΣΠΣ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theo...

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S = k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.

Multiplicative Bijections of Semigroups of Interval-valued Continuous Functions

We characterize all compact and Hausdorff spaces X which satisfy that for every multiplicative bijection φ on C(X, I), there exist a homeomorphism μ : X −→ X and a continuous map p : X −→ (0,+∞) such that φ(f)(x) = f(μ(x)) for every f ∈ C(X, I) and x ∈ X. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. 134 (2006), 1065-1075). Some related results on other semigroups of...

A characterization of triple semigroup of ternary functions and Demorgan triple semigroup of ternary functions

Multiplicative valued difference fields

The theory of valued difference fields (K,σ, v) depends on how the valuation v interacts with the automorphism σ. Two special cases have already been worked out the isometric case, where v(σ(x)) = v(x) for all x ∈ K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where v(σ(x)) > nv(x) for all x ∈ K× with v(x) > 0 and n ∈ N, has been worked out b...