Narrow Results

In this paper, several theorems of Macdonald [On certain varieties of groups, Math. Z.76 (1961) 270–282; On certain varieties of groups II, Math. Z.78 (1962) 175–188] on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some equations (see Eqs. (1.1)–(1.3)). We prove that each Lie ring in variety (1.1) is nilpotent of exponent at most n + 2; if L is a Lie ring in variety (1.2), then L² is nilpotent of exponent at most n + 1; and each Lie ring in variety (1.3) is solvable of length at most n + 1. We also discuss some varieties of solvable Lie rings and the varieties of Lie rings defined by the properties of subrings.

In this paper, we study one class of commutative operads, namely, the operads of multidimensional (hollow) cubes in Euclidean spaces and their generalization. We describe the varieties of universal algebras rational equivalent to the varieties of algebras over commutative operads.

In this paper, we first show that the variety ${𝒱}$ of posemigroups satisfying an identity $axy=axay$ is closed if it is ${𝒱}$-convex. Next, we show that inverse posemigroups satisfying an identity $x=xyx$ are absolutely closed. We also show that a convex variety of left [right] zero posemigroups is absolutely closed and finally $U^0$ is absolutely closed if so is $U$.