ON CONGRUENCE-COHERENT REES ALGEBRAS AND ALGEBRAS WITH AN OPERATOR
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 is called coherent if each of its subalgebras containing a class of some congruence on 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 with a nullary operation 0 is called weakly coherent if each of its subalgebras including the kernel of some congruence on is a union of classes of this congruence. An algebra with a nullary operation 0 is called locally coherent if each of its subalgebras including a class of some congruence on 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, ⟩ with one ternary operation 𝑑(𝑥, 𝑦,𝑧) and one unary operation 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, ⟩ 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.