Clones satisfying the term condition (Q798688)

An n-ary operation f satisfies the term condition (TC) if f and all its variants obtained by a permutation of the variables satisfy the biconditional \(f(x,u_ 1,...,u_{n-1})=f(x,v_ 1,...,v_{n-1})\) iff \(f(y,u_ 1,...,u_{n-1})=f(y,v_ 1,...,v_{n-1}). TC_ n(A)\) (resp. TC(A)) is the set of all n-ary (resp. all finitary) operations on A that satisfy TC. For \(k=\{0,1,...,k-1\} TC_ n(2)\) and \(TC_ n(3)\) are completely described; for \(k\geq 4\) bounds are given for \(TC_ n(k)\) showing that \((TC_ n(k))_ n\) is log-linear in n. A clone C of operations on a set A is a TC clone if \(C\subseteq TC(A)\). It is shown that there are countably many clones on a finite set; for finite A there are only finitely many maximal TC clones over A; the (two) maximal TC clones over a 3-element set are described. An algebra is TC if its clone is a TC clone; a TC variety is a variety of TC algebras; if C is a clone over A, Var(C) is the variety generated by the algebra (A,C). It is shown that for Var(C) to be TC it is not enough that C be a TC clone; a sufficient condition is that Var(C) be Hamiltonian (any subalgebra is a congruence class). These are scattered results among plenty of theorems and examples which make this paper a valuable source on the subject.