Optimal strong Mal'cev conditions for omitting type 1 in locally finite varieties. (Q2510719)

It is a recent discovery by M. Siggers that the property of omitting type 1 is equivalent to a strong Malcev condition for locally finite varieties. Siggers proved that the class of locally finite varieties omitting type 1 is definable within the class of locally finite varieties as the subclass of varieties with a 6-ary Taylor term. In this paper, the authors strengthen this result to obtain strong Malcev conditions for omitting type 1 using the generalization of the Hell-Nešetřil theorem that was proved by L. Barto, M. Kozik and T. Niven. The authors show that the class of locally finite varieties omitting type 1 is definable by idempotent, linear, strong Malcev conditions in the language with one 4-ary operation or with two 3-ary operations and this limit on the number of variables is sharp.