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Archimedean classes in an ordered semigroup. IV. (English) Zbl 0669.20048
Let S be an ordered semigroup, that is, a semigroup with a linear order \(\leq\) which is compatible with multiplication. For any \(x,y\in S\), set x\({\mathcal A}y\) iff for some natural numbers p, q, r and s, \(x^ p\leq y^ q\) and \(y^ r\leq x^ s\). It is well known that \({\mathcal A}\) is an equivalence relation called the archimedean equivalence on S. It is also known that each archimedean class C of S is a convex subsemigroup of S containing at most one idempotent. If C contains an idempotent, then every element of C is of finite order and C is called periodic; otherwise C is said to be torsion-free. In his previous papers [Parts I-III, ibid. 26(101), 218-251 (1976; Zbl 0338.06005, Zbl 0338.06006 and Zbl 0338.06007)] the author studied the behavior of the set product AB of two archimedean classes A and B of S depending on whether A and B are periodic or torsion-free and satisfy certain other related conditions. Since \({\mathcal A}\) in general is not a congruence, AB is not necessarily contained in a single archimedean class. In the present paper the author introduces the notion of a modified archimedean class of the given archimedean class C of S (this notion is defined separately for torsion- free and for periodic archimedean classes). For each pair of modified archimedean classes, their set product is contained in some unique modified archimedean class. This fact is used by the author for classifying set products of a finite number of archimedean classes of S.
Reviewer: S.M.Goberstein

MSC:
20M10 General structure theory for semigroups
06F05 Ordered semigroups and monoids
20M15 Mappings of semigroups
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References:
[1] T. Saitô: Regular elements in an ordered semigroup. Pacific J. Math. 13 (1963), 263-295. · Zbl 0113.25202
[2] T. Saitô: Archimedean classes in an ordered semigroup I. Czechoslovak Math. J. 26 (101) (1976), 218-238. · Zbl 0338.06005
[3] T. Saitô: Archimedean classes in an ordered semigroup II. Czechoslovak Math. J. 26 (101) (1976), 239-247. · Zbl 0338.06006
[4] T. Saitô: Archimedean classes in an ordered semigroup III. Czechoslovak Math. J. 26 (101) (1976), 248-251. · Zbl 0338.06007
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