On McKenzie’s method

Dedicated to Lászlo ́ Fuchs on the occasion of his 70th birthday This is an expository account of R. McKenzie’s recent refutation of the RS conjecture. We adopt the usual conventions: if S is an algebra then S denotes the universe of S, |S | denotes the cardinality of S, and |S|+ denotes the successor cardinal to |S|. For an algebra A, let Vsi(A) denote the class of all nontrivial subdirectly irreducible members of V(A), and (following McKenzie) define κ(A) = sup{|S|+: S ∈ Vsi(A)}. A long time ago R. Quackenbush asked [6] whether there exists a finite al-gebra A satisfying κ(A) = ω. In their book on tame congruence theory [2], D. Hobby and R. McKenzie conjectured that κ(A) ≥ ω implies κ(A) = ∞ for finite algebras A (the ‘RS conjecture’). But in 1993 McKenzie [3] dis-covered a method which allowed him to construct a finite algebra Aω with κ(Aω) = ω, and a variant of the method with which he could construct a finite algebra Aω1 of finite type with κ(Aω1) = ω1. Subsequently, McKenzie [4], M. Valeriote [7] and C. Latting (unpublished) used McKenzie’s second method to construct related examples. However, nowhere has the second method been explicitly described, that is, as a general method from which the cited examples are extracted as particular cases. It is our wish to give an explicit description of McKenzie’s second method, similar to McKenzie’s description of his first method [3, Sections 1-2]. To aid the reader, we begin the paper with something easier