A strong Mal'cev condition for locally finite varieties omitting the unary type
The main theorem claims that a finite algebra \(\mathbf A\) admits Taylor operations if and only if it admits an idempotent 6-ary operation satisfying the identities: \(\omega (x,x,x,x,y,y)=\omega (x,y,x,y,x,x)\) and \(\omega (y,y,x,x,x,x)=\omega (x,x,y,x,y,x)\). This result implies that a locally finite variety omits the unary type if an only if it has an idempotent operation \(\omega\) satisfying the identities above. The author mentions that ``this is of interest to combinatorialists as it is conjectured that a Constraint Satisfaction Problem defined by a core relational structure is polynomial time solvable exactly when a certain associated variety omits the unary type. Our result implies that the problem of deciding if a core relational structure meets this characterisation is itself in NP''.
scientific article; zbMATH DE number 5833769
scientific article