# zbMATH — the first resource for mathematics

On some varieties of weakly associative lattice groups. (English) Zbl 0870.06016
The notion of weakly associative lattice group (wal-group) as a generalization of $$l$$-group was introduced by the author [Acta Univ. Palack. Olomuc., Fac. Rerum Nat. 61, Math. 18, 5-20 (1979; Zbl 0436.06014)]. A wal-group is a structure $$(G,\wedge ,\vee ,+)$$ where $$(G,\wedge ,\vee)$$ is a weakly associative lattice [E. Fried, Acta Sci. Math. 31, 233-244 (1970; Zbl 0226.06005)], $$(G,+)$$ is a group and the addition $$+$$ is compatible with both operations $$\wedge$$ and $$\vee$$. In contrast to $$l$$-groups there exist also finite wal-groups. In the present paper the author studies properties of wal-ideals (i.e. kernels of wal-homomorphisms) and, especially, deals with the lattice $$\mathcal L_{{\operatorname {wal}}}$$ of all varieties of wal-groups. He shows that the class $$\mathcal R_{{\operatorname {wal}}}$$ of all representable wal-groups forms a variety incomparable with the variety $$\mathcal Ab_{{\operatorname {wal}}}$$ of all abelian wal-groups. In the end of the paper he gives a sketch of a diagram of the lattice $$\mathcal L_{{\operatorname {wal}}}.$$
Reviewer: R.Halaš (Olomouc)

##### MSC:
 06F25 Ordered rings, algebras, modules
##### Keywords:
wal-group; variety of wal-groups
##### References:
 [1] M. Anderson and T. Feil: Lattice-Ordered Groups (An Introduction). Reidel, Dordrecht-Boston-Lanaaster-Tokyo, 1988. · Zbl 0636.06008 [2] S. Burris and H. P. Sankappanavar: A Course in Universal Algebra. Springer-Verlag, New York-Heidelberg-Berlin, 1981. · Zbl 0478.08001 [3] E. Fried: A generalization of ordered algebraic systems. Acta Sci. Math. (Szeged) 31 (1970), 233-244. · Zbl 0226.06005 [4] A. M. W. Glass and W. Charles Holland (Eds.): Lattice-Ordered Groups (Advances and Techniques). Kluwer Acad. Publ., Dordrecht-Boston-London, 1989. · Zbl 0705.06001 [5] V. M. Kopytov: Lattice-Ordered Groups. Nauka, Moscow, 1984. · Zbl 0567.06011 [6] A. G. Kurosch: Lectures on General Algebra. Fizmatgiz, Moscow, 1962. [7] N. Ya. Medvedev: Varieties of Lattice-Ordered Groups. Altai Univ., Barnaul, 1987. · Zbl 0639.06014 [8] J. Rachůnek: Semi-ordered groups. Acta Univ. Palack. Olom., Fac. Rer. Nat. 61 (1979), 5-20. [9] J. Rachůnek: Solid subgroups of weakly associative lattice groups. Acta Univ. Palack. Olom., Fac. Rer. Nat. 105, Math. 31 (1992), 13-24. · Zbl 0776.06016 [10] J. Rachůnek: Groupes faiblement réticulés. Sém. de Structures Alg. Ordonnées 42, Univ. Paris VII, 1993, 11 pp. [11] H. Skala: Trellis theory. Alg. Univ. 1 (1971), 218-233. · Zbl 0242.06003 · doi:10.1007/BF02944982 [12] H. Skala: Trellis Theory. Memoirs AMS, Providence, 1972. · Zbl 0242.06004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.