On congruence-coherent Rees algebras and algebras with an operator

A. N. Lata

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 154-172.

Résumé

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.

Détail

Keywords: congruence lattice, coherence, weakly coherence, locally coherence, Rees algebra, Rees congruence, algebra with operators, unar with Mal’tsev operation, near-unanimity operation, weak near-unanimity operation.

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