Geodesic
Non-transitive generalizations of subdirect products of linearly ordered rings
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 591-603
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Weakly associative lattice rings (wal-rings) are non-transitive generalizations of lattice ordered rings (l-rings). As is known, the class of l-rings which are subdirect products of linearly ordered rings (i.e. the class of f-rings) plays an important role in the theory of l-rings. In the paper, the classes of wal-rings representable as subdirect products of to-rings and ao-rings (both being non-transitive generalizations of the class of f-rings) are characterized and the class of wal-rings having lattice ordered positive cones is described. Moreover, lexicographic products of weakly associative lattice groups are also studied here.
Weakly associative lattice rings (wal-rings) are non-transitive generalizations of lattice ordered rings (l-rings). As is known, the class of l-rings which are subdirect products of linearly ordered rings (i.e. the class of f-rings) plays an important role in the theory of l-rings. In the paper, the classes of wal-rings representable as subdirect products of to-rings and ao-rings (both being non-transitive generalizations of the class of f-rings) are characterized and the class of wal-rings having lattice ordered positive cones is described. Moreover, lexicographic products of weakly associative lattice groups are also studied here.
Classification : 06F15, 06F25, 13J25, 16W80
Keywords: weakly associative lattice ring; weakly associative lattice group; representable wal-ring 
@article{CMJ_2003_53_3_a7,
     author = {Rach\r{u}nek, Ji\v{r}{\'\i} and \v{S}alounov\'a, Dana},
     title = {Non-transitive generalizations of subdirect products of linearly ordered rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {591--603},
     year = {2003},
     volume = {53},
     number = {3},
     mrnumber = {2000055},
     zbl = {1080.06032},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a7/}
}
TY  - JOUR
AU  - Rachůnek, Jiří
AU  - Šalounová, Dana
TI  - Non-transitive generalizations of subdirect products of linearly ordered rings
JO  - Czechoslovak Mathematical Journal
PY  - 2003
SP  - 591
EP  - 603
VL  - 53
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a7/
LA  - en
ID  - CMJ_2003_53_3_a7
ER  - 
%0 Journal Article
%A Rachůnek, Jiří
%A Šalounová, Dana
%T Non-transitive generalizations of subdirect products of linearly ordered rings
%J Czechoslovak Mathematical Journal
%D 2003
%P 591-603
%V 53
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a7/
%G en
%F CMJ_2003_53_3_a7
Rachůnek, Jiří; Šalounová, Dana. Non-transitive generalizations of subdirect products of linearly ordered rings. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 591-603. http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a7/

[1] A.  Bigard, K. Keimel and S. Wolfenstein: Groupes et anneaux réticulés. Springer Verlag, Berlin-Heidelberg-New York, 1977. | MR

[2] S. Burris and H. P. Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York-Heidelberg-Berlin, 1981. | MR

[3] E.  Fried: Tournaments and non-associative lattices. Ann. Univ. Sci. Budapest, Sect. Math. 13 (1970), 151–164. | MR

[4] L. Fuchs: Partially Ordered Algebraic Systems. Mir, Moscow, 1965. (Russian) | MR

[5] V. M.  Kopytov: Lattice Ordered Groups. Nauka, Moscow, 1984. (Russian) | MR | Zbl

[6] V. M. Kopytov, N. Ya.  Medvedev: The Theory of Lattice Ordered Groups. Kluwer Acad. Publ., Dordrecht, 1994. | MR

[7] A. G.  Kurosch,: Lectures on General Algebra. Academia, Praha, 1977. (Czech)

[8] J.  Rachůnek: Solid subgroups of weakly associative lattice groups. Acta Univ. Palack. Olom. Fac. Rerum Natur. 105, Math. 31 (1992), 13–24. | MR

[9] J.  Rachůnek: Circular totally semi-ordered groups. Acta Univ. Palack. Olom. Fac. Rerum Natur. 114, Math. 33 (1994), 109–116. | MR

[10] J.  Rachůnek: On some varieties of weakly associative lattice groups. Czechoslovak Math. J. 46 (121) (1996), 231–240. | MR

[11] J.  Rachůnek: A weakly associative generalization of the variety of representable lattice ordered groups. Acta Univ. Palack. Olom. Fac. Rerum Natur., Math. 37 (1998), 107–112. | MR

[12] J.  Rachůnek: Weakly associative lattice groups with lattice ordered positive cones. In: Contrib. Gen. Alg. 11, Verlag Johannes Heyn, Klagenfurt, 1999, pp. 173–180. | MR

[13] D.  Šalounová: Weakly associative lattice rings. Acta Math. Inform. Univ. Ostraviensis 8 (2000), 75–87. | MR

[14] H.  Skala: Trellis theory. Algebra Universalis  1 (1971), 218–233. | DOI | MR | Zbl

[15] H.  Skala: Trellis Theory. Memoirs AMS, Providence, 1972. | MR | Zbl