Sums and products of intervals in ordered semigroups

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 I_x := {1, 2, . . ., x} for all x ∈ ℤ₊ and defined the function g : ℤ₊ → ℤ₊ by g(x):=max{ y∈ℤ+|Iy⊆Ix⋅Ix } 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.