The variety generated by \(\mathbb {A}(\mathcal {T})\) -- two counterexamples (Q2634704)

A variety \textbf{V} of algebras is said to \textit{have a finite residual bound} if there is a finite bound on the size of subdirectly irreducible algebras in \textbf{V}. \textit{R. E. Park}'s conjecture [Equational classes of non-associative ordered algebras. Los Angeles, CA: University of California (PhD Thesis) (1976)] says that every finitely generated variety with a finite residual bound is finitely based. The problem is still open but it was established for: congruence distributive varieties by \textit{K. A. Baker} [Adv. Math. 24, 207--243 (1977; Zbl 0356.08006)], congruence modular varieties by \textit{R. McKenzie} [Algebra Univers. 24, No. 3, 224--250 (1987; Zbl 0648.08006)] and congruence \(\wedge\)-semidistributive varieties by \textit{R. Willard} [J. Symb. Log. 65, No. 1, 187--200 (2000; Zbl 0973.08004)]. This paper gives the first example of a variety \textbf{V} generated by a single finite algebra \(A\) of a finite type, such that \textbf{V} has finite residual bound and is congruence \(\wedge\)-semidistributive, but has neither definable principal congruences nor bounded Maltsev depth. This way two questions asked by \textit{R. Willard} [Algebra Univers. 45, No. 2--3, 335--344 (2001; Zbl 0980.08003); Contrib. Gen. Algebra 15, 199--206 (2004; Zbl 1068.08005)] are answered in the negative. The author uses as \(A\) the algebra associated to a Turing machine which was constructed by \textit{R. McKenzie} in [Int. J. Algebra Comput. 6, No. 1, 29--48 (1996; Zbl 0844.08010)]. However, the paper is self-contained.