Left and right nilpotence degree are independent (Q2759841)

Let \({\mathcal A}\) be a universal algebra of type \(\tau\) and \(\alpha,\beta\in\text{ Con}{\mathcal A}\). Then the commutator \([\alpha,\beta]\) of \(\alpha\) and \(\beta\) is the least congruence \(\gamma\) on \({\mathcal A}\) such that if \((a,b)\in\alpha\), \(n\) is a positive integer, \((a_1,b_1),\dots,(a_n,b_n)\in\beta\), \(t\) is an \((n+1)\)-ary term of type \(\tau\) and \((t(a,a_1,\dots,a_n),t(a,b_1,\dots,b_n))\in\gamma\) then \((t(b,a_1,\dots,a_n),t(b,b_1,\dots,b_n))\in\gamma\). The left nilpotence degree of \(\alpha\) is the smallest positive integer \(n\) (if such an integer exists) such that \([\alpha,\dots[\alpha,[\alpha,\alpha]]\dots]\) (with \(\alpha\) occurring \(n+1\) times) is the least congruence on \({\mathcal A}\). The right nilpotence degree of \(\alpha\) is defined dually. It is proved that for arbitrary integers \(n,m>1\) there exists a finite algebra having a congruence with left nilpotence degree \(n\) and right nilpotence degree \(m\). This solves a problem posed by K. A. Kearnes.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].