Gardner, B. J.; Parmenter, M. M. Directoid groups. (English) Zbl 1174.06340 Czech. Math. J. 58, No. 3, 669-681 (2008). Summary: We continue the study of directoid groups, directed abelian groups equipped with an extra binary operation which assigns an upper bound to each ordered pair subject to some natural restrictions. The class of all such structures can to some extent be viewed as an equationally defined substitute for the class of (2-torsion-free) directed abelian groups. We explore the relationship between the two associated categories, and some aspects of ideals of directoid groups. MSC: 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces Keywords:directed abelian group; variety × Cite Format Result Cite Review PDF Full Text: DOI EuDML Link References: [1] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et anneaux réticulés. Berlin etc., Springer, 1977. · Zbl 0384.06022 [2] L. Fuchs: Absolutes in partially ordered groups. Kon. Nederl. Akad. Wetensch. Proc. Amsterdam 52 (1949), 251–255. · Zbl 0033.10002 [3] L. Fuchs: Partially ordered algebraic systems. Oxford-London-New York-Paris, Pergamon Press, 1963. · Zbl 0137.02001 [4] B. J. Gardner and M. M. Parmenter: Directoids and directed groups. Algebra Univ. 33 (1995), 254–273. · Zbl 0832.06005 · doi:10.1007/BF01190937 [5] T. E. Hall: Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc. 40 (1989), 59–77. · Zbl 0666.20028 · doi:10.1017/S000497270000349X [6] P. J. Higgins: Groups with multiple operators. Proc. London Math. Soc. 6 (1956), 366–416. · Zbl 0073.01704 · doi:10.1112/plms/s3-6.3.366 [7] P. Jaffard: Un contre-exemple concernant les groupes de divisibilité. C.R. Acad. Sci. Paris 243 (1956), 1264–1266. · Zbl 0071.25405 [8] J. Jakubík: On directed groups with additional operations. Math. Bohem. 118 (1993), 11–17. · Zbl 0799.06027 [9] J. Ježek and R. Quackenbush: Directoids: algebraic models of up-directed sets. Algebra Univ. 27 (1990), 49–69. · Zbl 0699.08002 · doi:10.1007/BF01190253 [10] V. M. Kopytov and Z. I. Dimitrov: On directed groups. Siberian Math. J. 30 (1989), 895–902. · Zbl 0714.06007 · doi:10.1007/BF00970912 [11] A. G. Kurosh: Lectures on general algebra. New York, Chelsea, 1963. · Zbl 0121.25901 [12] K. Leutola and J. Nieminen: Posets and generalized lattices. Algebra Univ. 16 (1983), 344–354. · Zbl 0514.06003 · doi:10.1007/BF01191789 [13] D. B. McAlister: On multilattice groups. Proc. Cambridge Phil. Soc. 61 (1965), 621–638. · Zbl 0135.06203 · doi:10.1017/S0305004100038962 [14] J. Nieminen: On distributive and modular {\(\chi\)}-lattices. Yokohama Math. J. 31 (1983), 13–20. · Zbl 0532.06002 [15] V. Snášel: {\(\lambda\)}-lattices. Math. Bohem. 122 (1997), 267–272. · Zbl 0897.06003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.