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Structure and laws of the Scrimger varieties of lattice-ordered groups. (English) Zbl 0624.06023
Algebra and order, Proc. 1st Int. Symp., Luminy-Marseilles/France 1984, Res. Expo. Math. 14, 71-81 (1986).
[For the entire collection see Zbl 0601.00004.]
Let p be a prime number. \(G=D(T,H,K,\mu)\) is called a p-dent if (i) T is a non-zero Abelian o-group (ii) H, the lex kernel of G, is the cardinal sum of p copies of T (iii) \(K=G/H\) is an abelian o-group and (iv) \(\mu\) : \(G\to {\mathbb{Z}}/(p)\) is an epimorphism satisfying for all \(h\in H\), \(g\in G\), \((g^{-1}hg)(i)=h(i-\mu (g))\) (0\(\leq i\leq p-1)\). It is shown that any p-dent can be embedded in a split p-dent (the key step being that G can be embedded in \(G^*=D(T^*,H^*,K^*,\mu)\) where \(T^*\), \(H^*\) and \(K^*\) are the divisible closures of T, H and K respectively. It is further proved that any split p-dent (and hence any p-dent) belongs to \({\mathfrak S}p\) the Scrimger p-cover of the variety of abelian lattice-ordered groups, and that \({\mathfrak S}p\) is defined by the laws \[ (i)\quad x^ py^ p=y^ px^ p\quad and\quad (ii)\quad e=| [x,y][x,y]^ z...[x,y]^{z^{p-1}}| \quad \wedge \quad | [u,v][u,v]^{z^{-1}}|. \] The proofs are straightforward and much easier than those needed to obtain the deeper results by S. A. Gurchenkov [Algebra Logika 23, 27-47 (1984; Zbl 0544.06013)] which were subsequently proved by the authors and A. H. Mekler [Algebra Univers. 23, 196-214 (1986; Zbl 0598.06008)].
Reviewer: A.M.W.Glass

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20F60 Ordered groups (group-theoretic aspects)
20E10 Quasivarieties and varieties of groups
08B15 Lattices of varieties
06B20 Varieties of lattices