О КОНГРУЭНЦ–КОГЕРЕНТНЫХ АЛГЕБРАХ РИСА И АЛГЕБРАХ С ОПЕРАТОРОМ

Abstract

The paper contains a classification of congruence-coherent Rees algebras and algebras with an operator. The concept of coherence was introduced by D.Geiger. An algebra A is called coherent if each of its subalgebras containing a class of some congruence on A is a union of such classes. In Section 3 conditions for the absence of congruence-coherence property for algebras having proper subalgebras are found. Necessary condition of congruence-coherence for Rees algebras are obtained. Sufficient condition of congruence-coherence for algebras with an operator are obtained. In this section we give a complete classification of congruence-coherent unars. In Section 4 some modification of the congruence-coherent is considered. The concept of weak and locally coherence was introduced by I.Chajda. An algebra A with a nullary operation 0 is called weakly coherent if each of its subalgebras including the kernel of some congruence on A is a union of classes of this congruence. An algebra A with a nullary operation 0 is called locally coherent if each of its subalgebras including a class of some congruence on A also includes a class the kernel of this congruence. Section 4 is devoted to proving sufficient conditions for algebras with an operator being weakly and locally coherent. In Section 5 deals with algebras ⟨ A,d,f ⟩ with one ternary operation d ( x,y,z ) and one unary operation f acting as endomorphism with respect to the operation d ( x,y,z ). Ternary operation d ( x,y,z ) was defined according to the approach offered by V.K. Kartashov. Necessary and sufficient conditions of congruence-coherent for algebras ⟨ A,d,f ⟩ are obtained. Also, necessary and sufficient conditions of weakly and locally coherent for algebras ⟨ A,d,f, 0⟩ with nullary operation 0 for which f (0) = 0 are obtained.