Sums and products of intervals in ordered semigroups

Abstract

We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b. The multiplicative version of the above example is shown too. The product of open intervals in the ordered ring of all integers (denoted by ℤ) is also investigated. Let Ix := {1, 2, . . ., x} for all x ∈ ℤ+ and defined the function g : ℤ+ → ℤ+ by g ( x ) : = max { y ∈ ℤ + | I y ⊆ I x ⋅ I x } g\left( x \right): = \max \left\{ {y \in {\mathbb{Z}_ + }|{I_y} \subseteq {I_x} \cdot {I_x}} \right\} for all x ∈ ℤ+. We give the function g implicitly using the famous Theorem of Chebishev. Finally, we formulate some questions concerning the above topics.

Existence and uniqueness theorems for functional equations

“…Property (2) can be easily deduced from property (1), although in [7] can be found an example of dense Abelian semigroup which has property (2) without property (1). Similarly, property (4) can be easily deduced from property (3), although in [7] can also be found an example for dense Abelian semigroup which has property (4) without property (3).…”

Short remark to the Rimán’s Theorem

“…Let a : R → R is a noncontinuous additive function. As it is wellknown that the graph of a is dense in R 2 (with respect to usual topology on R 2 ), but the restriction of the function a to the set Q (where Q denotes the set of all rationals) is continuous with respect to the topology on the set Q(+) defined above, and the usual topology on the real line [11].…”