Subtraces and shadow chains (Q5932481)

For types, the following partial order is considered: \(1<2<3\) and \(1<5<4<3\) (where 2 and 5 or 2 and 4 are incomparable). Theorem. Let \(B\) be a finite algebra in a variety \(\mathcal V\) and suppose that type \(\{ S(B)\}\) omits all types strictly less than a given type \(i\). Then \(i\in \text{typ}(B)\) implies that \(i\) belongs to types of the free algebra of \(\mathcal V\) with two free generators. As a corollary the author obtains an answer to a question posed by R. N. McKenzie: Let \({\mathcal V}(A)\) be the variety generated by the finite algebra \(\mathcal A\). If \({\mathcal V}(A)\) omits 1 from its typeset, then it is decidable if 5 occurs in type \(\{ {\mathcal V} (A)\}\).