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Pretoria

By Elemer E Rosinger

Abstract

Two concepts of being Archimedean are defined for arbitrary categories. 1. The case of usual semigroups For convenience, let us recall two versions of concepts of being Archimedean in the usual case of algebraic structures. In this regard, a sufficiently general setup is as follows. Let (E, +, ≤) be a partially ordered semigroup, thus we have satisfied (1.1) x, y ∈ E+ = ⇒ x + y ∈ E+ where E+ = {x ∈ E | x ≥ 0}. A first intuitive version of the Archimedean condition, suggested in case ≤ is a linear or total order on E, is (1.2) ∃ u ∈ E+: ∀ x ∈ E: ∃ n ∈ N: nu ≥ x Here is other formulation used in the literature when ≤ is an arbitrary partial order on E 1 (1.3) ∀ x ∈ E+: x = 0 ⇐⇒ ∃ y ∈ E+: ∀ n ∈ N: nx ≤ y where clearly the implication ”=⇒ ” is trivial, and which condition is thus equivalent with (1.4) ∀ x ∈ E+: Nx is bounded above = ⇒ x =

Topics: category C 2, associated with C, H & s, namely, the category whose class of objects is the class of morphisms of C, w
OAI identifier: oai:CiteSeerX.psu:10.1.1.313.6030
Provided by: CiteSeerX

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