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**Varieties having the congruence extension property.**
*(English)*
Zbl 0861.08006

Recall that a tolerance on an algebra \(A\) is a reflexive, symmetric binary relation on \(A\) which is a subalgebra of the direct power \(A\times A\). The set of all tolerances on \(A\) forms a complete lattice, hence, for any \(a,b\in A\) there exists the least tolerance containing the pair \(\langle a,b\rangle\). This tolerance is denoted by \(T_A(a,b)\) and is called the principal tolerance generated by \(\langle a,b\rangle\). An algebra \(A\) is called principal tolerance trivial if \(T_A(a,b)\) is a congruence on \(A\) for any \(a\), \(b\) of \(A\). A variety \(\mathcal V\) is principal tolerance trivial if every \(A\) of \(\mathcal V\) has this property.

A variety \(\mathcal V\) has the Congruence Extension Property, briefly CEP, if for each \(A\in {\mathcal V}\), every subalgebra \(B\) of \(A\) and each \(\theta \in \text{Con }B\) there exists \(\Phi \in \text{Con }A\) with \(\Phi|B=\theta\); \(\Phi\) is called the extension of \(\theta\). It is known [A. Day, Algebra Univers. 1, 234-235 (1971; Zbl 0228.08001)] that varieties having CEP cannot be characterized by a Mal’tsev condition. A paper of the author [Acta Sci. Math. 56, 19-21 (1992; Zbl 0768.08005)] contains another condition using term functions characterizing varieties which are congruence-permutable and have CEP.

The aim of the present paper is to generalize this also to non-permutable varieties and to varieties having trivial principal tolerances.

A variety \(\mathcal V\) has the Congruence Extension Property, briefly CEP, if for each \(A\in {\mathcal V}\), every subalgebra \(B\) of \(A\) and each \(\theta \in \text{Con }B\) there exists \(\Phi \in \text{Con }A\) with \(\Phi|B=\theta\); \(\Phi\) is called the extension of \(\theta\). It is known [A. Day, Algebra Univers. 1, 234-235 (1971; Zbl 0228.08001)] that varieties having CEP cannot be characterized by a Mal’tsev condition. A paper of the author [Acta Sci. Math. 56, 19-21 (1992; Zbl 0768.08005)] contains another condition using term functions characterizing varieties which are congruence-permutable and have CEP.

The aim of the present paper is to generalize this also to non-permutable varieties and to varieties having trivial principal tolerances.

Reviewer: Leonid Matveevich Martynov (Omsk)

### MSC:

08B05 | Equational logic, Mal’tsev conditions |

### References:

[1] | Day A.: A note on the Congruence Extension Property. Algebra Univ. 1 (1971), 234-235. · Zbl 0228.08001 · doi:10.1007/BF02944983 |

[2] | Chajda I.: Algebras and varieties satisfying the Congruence Extension Property. Acta Sci. Math. (Szeged) 56 (1992), 19-21. · Zbl 0768.08005 |

[3] | Chajda I.: Tolerance trivial algebras and varieties. Acta Sci. (Szeged) 46 (1983), 35-40. · Zbl 0534.08001 |

[4] | Chajda I.: Algebraic Theory of Tolerance Relations. Monograph Series of Palacky University, Olomouc, 1991. · Zbl 0747.08001 |

[5] | Niederle J.: Conditions for trivial principal tolerances. Arch. Math. (Brno) 19 (1983), 145-152. · Zbl 0538.08002 |

[6] | Niederle J.: Conditions for transitive principal tolerances. Czech. Math. J. 39 (1989), 380-381. · Zbl 0686.08003 |

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