Narrow Results

Let be a variety of the form where is a finite subdirectly irreducible algebra. We show that if is naturally dualizable (in the sense of D. M. Clark and B. A. Davey, i.e. with respect to the discrete topology) then the variety determined by all normal identities of (the so called nilpotent shift of ) is also naturally dualizable. We give a finite algebra and a relational system , constructed explicitly from the system for , such that and dualizes .

P-Compatibility is a hereditary property of identities which generalizes the properties of normality and externality of identities. Chajda characterized the normalization of a variety by an algebraic construction called a choice algebra. In this paper, we generalize this characterization to the least P-compatible variety P(V) determined by a variety V for any partition P using P-choice algebras. We also study the clone of (strongly) P-compatible n-ary terms of a variety V, and relate identities of this clone to (strongly) P-compatible hyperidentities of the variety V.

In this paper, we investigate essentially n-ary term operations of nilpotent extensions of algebras. We detect the connection between term operations of an original algebra and its nilpotent extensions. This structural point of view easily leads to the conclusion that the number of distinct essentially n-ary term operations of a proper algebraic nilpotent extension 𝔄 of an algebra ℑ is given by the formula