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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)

06F25 Ordered rings, algebras, modules
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