On the theory of varieties of lattice ordered groups (Q1822617)

Let the natural number \(n=p_ 1^{n_ 1}p_ 2^{n_ 2}...p_ r^{n_ r}\) be the product of prime numbers, where \(p_ 1,p_ 2,...,p_ r\) are distinct prime numbers, \(n_ i\geq 1\) \((i=1,2,...,r)\), \(\bar n=n_ 1+n_ 2+...+n_ r+1\) and \({\mathfrak L}_ n\) be the \(\ell\)- variety defined by the law \([x^ n,y^ n]=e\). In this paper the following main results are proved. 1) \({\mathfrak L}_ n\subseteq ({\mathfrak A}_{\ell})^{\bar n}\), where \({\mathfrak A}_{\ell}\) is the \(\ell\)- variety of all abelian \(\ell\)-groups (Theorem 1). 2) Let \({\mathfrak N}\) be an \(\ell\)-variety and every linearly ordered group from the \(\ell\)- variety \({\mathfrak N}\) is abelian. Then there exists a natural number \(n=n({\mathfrak N})\) such that \({\mathfrak N}\subseteq {\mathfrak L}_ n\) (Theorem 2). 3) The lattice of all \(\ell\)-subvarieties of the \(\ell\)-variety \({\mathfrak L}_ n\wedge ({\mathfrak A}_{\ell})^ 2\) is described and it is proved: a) every \(\ell\)-variety \({\mathfrak L}\subseteq {\mathfrak L}_ n\wedge ({\mathfrak A}_{\ell})^ 2\) has a finite basis of identities; b) if the \(\ell\)-variety \({\mathfrak L}\) has finite basis rank, then the lattice of all \(\ell\)-subvarieties of \({\mathfrak L}\) is finite (Theorems 6.7). 4) An \(\ell\)-variety of nilpotent \(\ell\)-groups of nilpotency class 3 with finite axiomatic rank and without independent basis of identities is constructed (Theorem 8). 5) The existence of linearly ordered nilpotent groups with the property \(var_{\ell}G\neq var_{\ell}G^*\), where \(G^*\) is the Malcev completion of the nilpotent group G, is established (Theorem 9).