The \(S\)-classification of idempotent algebras with a finite support (Q1380284)

Assume that \(k\) is a positive integer, \(E_k= \{0,1,\dots,k- 1\}\), \(P_k\) is the set of all finite-place operations on \(E_k\), \(\langle E_k; F\rangle\) is the algebra with the support \(E_k\) and the set of operations \(F\). Assume that \(\pi\) is a substitution on \(E_k\) and \(f(x_1,\dots, x_n)\in P_k\). The operation \(\pi^{-1}(f(\pi(x_1),\dots\), \(\pi(x_n)))\) is called dual to \(f\) relative to the substitution \(\pi\). If \(F\subseteq P_k\), then \([F]_S\) will denote the clone that is generated by all possible operations dual to operations from \(F\). We will call algebras \(\langle E_k;F\rangle\) and \(\langle E_k; G\rangle S\)-equivalent if \([F]_S= [G]_S\). A classification of some idempotent algebras relative to the \(S\)-equivalence was obtained by the author in earlier papers: for \(k\geq 3\) algebras with the complete symmetric group of automorphisms (homogeneous algebras) and for \(k\geq 4\) algebras with the alternating group of automorphisms. Fifty-two algebras of finite type were found by \textit{Nguyen Van Hoa} [Diskretn. Mat. 4, No. 4, 82-95 (1992; Zbl 0801.03018)] for \(k=3\); it was proved there that any algebra \(\langle E_3;F\rangle\) is \(S\)-equivalent to one of these algebras. A finite set of algebras of finite type was constructed by \textit{Nguyen Van Hoa} [Diskretn. Mat. 5, No. 4, 87-108 (1993; Zbl 0813.03016)] for any \(k\), \(k\geq 4\), to solve a similar problem for the algebras \(\langle E_k;F\rangle\) for which the set \(F\) contains no essentially many-place operations that take all the \(k\) values (Slupecki algebras). It was also shown there that for \(k\geq 5\), every algebra \(\langle E_k;F\rangle\) is either \(S\)-equivalent to the algebra \(\langle E_k; P_k\rangle\), idempotent, or is a Slupecki algebra. For \(k= 4\), one has to add to these possibilities algebras of quasilinear operations and algebras whose automorphism group coincides with Klein's four-group. Therefore, to construct the \(S\)-classification of algebras \(\langle E_k; F\rangle\) for \(k\geq 4\), one has to classify the algebras that are not Slupecki algebras, and, first of all, idempotent algebras. Conceptually and technically, this part of the \(S\)-classification is the most difficult. We will characterize all classes of \(S\)-equivalent idempotent algebras (for any \(k\), \(k\geq 3\)), and classes of \(S\)-equivalent algebras that are not Slupecki algebras (for \(k< 5\)). We will describe these classes in terms of relations.