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 $\langle A, d, f \rangle$ 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 $\langle A, d, f \rangle$ are obtained.
Also, necessary and sufficient conditions of weakly and locally coherent for algebras $\langle A, d, f, 0 \rangle$ with nullary operation $0$ for which $f(0)=0$ are obtained.
Bibliography: 33 titles.
