The Congruence Variety of Metaabelian Groups Is Not Self–dual
نویسنده
چکیده
A lattice identity is given such that it holds but its dual fails in the normal subgroup lattices of metaabelian groups. Thus the congruence variety of metaabelian groups is not self-dual; this is the first example for a modular congruence variety which is not self-dual. For a ring R with unit element let L(R) denote the class of lattices embeddable in submodule lattices of R-modules. Then HL(R), the variety generated by L(R), is a self-dual congruence variety by Hutchinson [6, Thm. 7, and 5]. On the other hand, non-modular congruence varieties need not be self-dual by Day and Freese [2]. The HL(R) have been the only known congruence varieties for a long time, leading to the impression that the congruence variety of Abelian groups, alias HL(Z), could be the largest modular congruence variety. This picture was refuted in two steps. First, an unpublished work of Kiss and Pálfy [7] showed that the congruence lattice of a certain metaabelian group cannot be embedded in the congruence lattice of any Abelian group. Developing these ideas further, Pálfy and Szabó [8, 9] have recently shown that the congruence variety of certain group varieties are not subvarieties of HL(Z). This leads to the problem whether every modular congruence variety is self-dual, cf. Pálfy and Szabó [9, Problem 4.2] for a slightly different formulation. The aim of the present paper is to give a negative solution. For a variety V let Con(V ) denote the congruence variety of V , i.e., the lattice variety generated by the congruence lattices of all algebras in V . Let M be the variety of metaabelian groups. M is defined by the identity [x, y]z = z[x, y] where [x, y] = x−1y−1xy. By the elementary properties of the commutator (cf., e.g., Received March 11, 1993. 1980 Mathematics Subject Classification (1991 Revision). Primary 08B10; Secondary 06C99.
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