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Directoids and directed groups. (English) Zbl 0832.06005
J. Ježek and R. Quackenbush [ibid. 27, No. 1, 49-69 (1990; Zbl 0699.08002)] have introduced directoids as groupoids corresponding exactly to up-directed posets (viz $$a \leq b$$ iff $$ab = b$$). In this paper, this concept is further investigated, both in its own and in combination with an abelian group structure.
Reviewer: J.Rosický (Brno)

##### MSC:
 06A99 Ordered sets 20N02 Sets with a single binary operation (groupoids) 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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##### References:
 [1] Dubreil-Jacotin, M. L., Lesieur, L. andCroisot, R.,LeÇons sur la théorie des treillis des structures algébriques ordonnées et des treillis géométriques, Paris, Gauthier-Villars, 1953. [2] Fuchs, L.,Absolutes in partially ordered groups, Kon. Nederl. Akad. Wetensch. Proc. Sec. Sci.52 (1949), 251-255. · Zbl 0033.10002 [3] Fuchs, L.,Partially ordered algebraic systems, Oxford-London-New York-Paris, Pergamon Press, 1963. · Zbl 0137.02001 [4] Gardner, B. J.,Radical decompositions of idempotent algebras, J. Austral. Math. Soc. (Ser. A)36 (1984), 213-236. · Zbl 0541.08001 · doi:10.1017/S1446788700024666 [5] Howie, J. M.,An introduction to semigroup theory, New York, Academic Press, 1976. · Zbl 0355.20056 [6] Jaffard, P.,Contribution à l’étude des groupes ordonnés, J. Math. Pures Appl.32 (1953), 203-280. · Zbl 0051.01303 [7] Jaffard, P.,Un contre-exemple concernant les groupes de divisibilité, C.R. Acad. Sci. Paris243 (1956), 1264-1268. · Zbl 0071.25405 [8] Je?ek, J. andQuackenbush, R.,Directoids: algebraic models of up-directed sets, Algebra Univ.27 (1990), 49-69. · Zbl 0699.08002 · doi:10.1007/BF01190253 [9] Mott, J. L., Groups of divisibility: a unifying concept for integral domains and partially ordered groups, pp. 80-104 ofLattice-ordered groups, Advances and techniques, Dordrecht-Boston-London, Kluwer Academic Publishers, 1989. [10] Ruedin, J.,Equivalences de Green et demi-treillis images homomorphes maximales d’un groupoide distributif, C.R. Acad. Sci. Paris Sér. A264 (1967), 429-432. · Zbl 0183.02201 [11] Shevrin, L. N. andMartynov, L. M.,On attainable classes of algebras, Siberian Math. J.12 (1971), 1363-1381. · Zbl 0233.08004 [12] Tamura, T.,Attainability of systems of identities on semigroups, J. Algebra3 (1966), 261-276. · Zbl 0146.02702 · doi:10.1016/0021-8693(66)90001-9
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