# Article

Full entry | PDF   (0.2 MB)
Keywords:
directed abelian group; variety
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.
References:
[1] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et anneaux réticulés. Berlin etc., Springer (1977). MR 0552653 | Zbl 0384.06022
[2] Fuchs, L.: Absolutes in partially ordered groups. Kon. Nederl. Akad. Wetensch. Proc. Amsterdam 52 (1949), 251-255. MR 0030526 | Zbl 0033.10002
[3] Fuchs, L.: Partially ordered algebraic systems. Oxford-London-New York-Paris, Pergamon Press (1963). MR 0171864 | Zbl 0137.02001
[4] Gardner, B. J., Parmenter, M. M.: Directoids and directed groups. Algebra Univ. 33 (1995), 254-273. DOI 10.1007/BF01190937 | MR 1318990 | Zbl 0832.06005
[5] Hall, T. E.: Identities for existence varieties of regular semigroups. Bull. Austral. Math. Soc. 40 (1989), 59-77. DOI 10.1017/S000497270000349X | MR 1020841 | Zbl 0666.20028
[6] Higgins, P. J.: Groups with multiple operators. Proc. London Math. Soc. 6 (1956), 366-416. MR 0082492 | Zbl 0073.01704
[7] Jaffard, P.: Un contre-exemple concernant les groupes de divisibilité. C.R. Acad. Sci. Paris 243 (1956), 1264-1266. MR 0086050
[8] Jakubík, J.: On directed groups with additional operations. Math. Bohem. 118 (1993), 11-17. MR 1213828
[9] Ježek, J., Quackenbush, R.: Directoids: algebraic models of up-directed sets. Algebra Univ. 27 (1990), 49-69. DOI 10.1007/BF01190253 | MR 1025835
[10] Kopytov, V. M., Dimitrov, Z. I.: On directed groups. Siberian Math. J. 30 (1989), 895-902. DOI 10.1007/BF00970912 | MR 1043436 | Zbl 0714.06007
[11] Kurosh, A. G.: Lectures on general algebra. New York, Chelsea (1963). MR 0158000
[12] Leutola, K., Nieminen, J.: Posets and generalized lattices. Algebra Univ. 16 (1983), 344-354. MR 0695054 | Zbl 0514.06003
[13] McAlister, D. B.: On multilattice groups. Proc. Cambridge Phil. Soc. 61 (1965), 621-638. MR 0175819 | Zbl 0135.06203
[14] Nieminen, J.: On distributive and modular $\chi$-lattices. Yokohama Math. J. 31 (1983), 13-20. MR 0734154 | Zbl 0532.06002
[15] Snášel, V.: $\lambda$-lattices. Math. Bohem. 122 (1997), 267-272. MR 1600648

Partner of