Hyperidentities of QZ-algebras (Q922570)

The symbol \({\mathfrak P}^*_ A\) denotes the algebra \(<P_ A;\zeta,\tau,\Delta,*>\) of type \(<1,1,1,2>\) where \(P_ A\) is the set of all operations on A and \((\zeta f)(x_ 1,...,x_ n)=f(x_ 2,...,x_ n,x_ 1)\), \((\tau f)(x_ 1,...,x_ n)=f(x_ 2,x_ 1,x_ 3,...,x_ n)\), \((\Delta f)(x_ 1,...,x_ n)=f(x_ 1,x_ 1,x_ 2,...,x_{n- 1})\), \((f*g)(x_ 1,...,x_{n+m-1})=f(g(x_ 1,..,x_ m),x_{m+1},...,x_{n+m-1})\). \({\mathfrak K}_ D\) denotes the subalgebra of \({\mathfrak P}^*_ A\) formed by operations whose values belong to a fixed subset D of A. An algebra \(<A;F>\) is called a QZ-algebra whenever the subalgebra of \({\mathfrak P}^*_ A\) generated by \(<A;F>\) is a union of some subalgebras \({\mathfrak K}_ D\), \(D\subseteq A\). An algebra \(<A;F>\) is called primal whenever A is finite and \(<A;F>\) generates \({\mathfrak P}^*_ A\). Primal algebras were characterized by systems of hyperidentities, disjunctions of which are false on these algebras. The paper gives similar criteria for 3-element QZ-algebras.