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The pre-clone of a variety

Denecke K., Wismath S.

Demonstratio Mathematica

2005

Vol. 38, nr 4

799--810

Clones are sets of operations on a given base set which are closed under superposition of operations and which contain all the projection operations. A clone can be regarded as a heterogeneous or multi-based algebra, consisting of universe sets of n-ary operations, for n > 1, with the superposition operations Snm, for n,m > 1. Such an algebra is called a Menger system. For a fixed n, a set of n-ary operations with the (n + 1)-ary superposition operation Sn forms a homogeneous algebra called a Menger algebra of rank n. These structures can also be defined on sets of term operations of a fixed type r. The set of all terms of type r which contain at least one operation symbol forms such a clone-like structure which we call the pre-clone of type r. We determine a generating system for this pre-clone. A similar structure can be formed using the terms with respect to a variety, giving the pre-clone of a variety. The normalization of a variety is the model class of all identities s~~t of the variety for which both s and t contain at least one operation symbol. We show that identities in the pre-clone of the normalization of a variety correspond to normal hyperidentities in the normalization.