Archimedean Type Conditions in Categories
نویسنده
چکیده
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 another formulation used in the literature when ≤ is an arbitrary partial order on E, namely
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