Clones of algebras with parallelogram terms (Q2520769)

Let \(A\) be a set and \(k\) a positive integer. A finitary operation \(f\) on \(A\) is compatible with a \(k\)-ary relation \(R\subseteq A^k\) if \(R\) is a subuniverse of the algebra \(\langle A;f\rangle^k\). This defines a Galois connection between finitary operations and relations on \(A\). For finite \(A\), the Galois closed subsets of operations are just the clones, i. e., the collections of finitary operations on \(A\) that contain the projections and are closed under composition. They are in a one-to-one correspondence with the Galois closed sets of relations, which are called finitary relational clones. Let \({\mathcal R}\) be such a finitary relational clone. The so-called ``critical'' relations, which are the directly indecomposable and completely \(\cap\)-irreducible members of \({\mathcal R}\), play a crucial role for generating \({\mathcal R}\).NEWLINENEWLINENEWLINENEWLINELet \(m,n\) be positive integers with \(m+n=k\), then an ``\((m,n)\)-parallelogram'' in \(A^k\) consists -- up to permutation of coordinates -- of four elements of the form \(\mathbf{ac},\mathbf{ad},\mathbf {bc}\) and \(\mathbf {bd}\) where \(\mathbf a,\mathbf b\in A^m\) and \(\mathbf c,\mathbf d\in A^n\). A relation \(R\subseteq A^k\) is said to satisfy the ``\((m,n)\)-parallelogram property'' if whenever it contains three vertices of an \((m,n)\)-parallelogram, then it contains the fourth. Now let \(\mathbf A\) be an algebra (with base set \(A\)) in a congruence modular variety. The first significant result of the paper is a structure theorem for the critical compatible relations of \(\mathbf A\) that satisfy the \((1,k-1)\)-parallelogram property.NEWLINENEWLINENEWLINENEWLINEFurthermore, the authors define ``\(k\)-ary parallelogram terms'', which are used to formulate a Maltsev condition for varieties. It is proved that this Maltsev condition is weaker than the one defining the class of congruence permutable varieties and also weaker than the one defining the class of varieties with a \(k\)-near unanimity term (but it is stronger than the one defining the class of congruence modular varieties).NEWLINENEWLINENEWLINENEWLINEThe main result of the paper is the following theorem: If \(\mathbf A\) is a finite algebra with a \(k\)-parallelogram term (\(k>1\)) such that \(\mathbf A\) generates a residually small variety, then the finitary relational clone of compatible relations of \(\mathbf A\) is generated by relations of arity \(\leq c\), where the constant \(c\) depends only on \(k\) and the size of the base set \(A\) of \(\mathbf A\). As a consequence, for a fixed integer \(k>1\) and a fixed finite set \(A\) there are only finitely many clones of algebras on \(A\) that have a \(k\)-parallelogram term and generate a residually small variety.NEWLINENEWLINENEWLINENEWLINEFinally, the reviewer wants to remark that this excellent paper is written in a very clear style, the proofs of the deep results are partly technical and quite sophisticated.