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**Algebraic theory of tolerance relations.**
*(English)*
Zbl 0747.08001

Olomouc: Univerzita Palackého Olomouc. 115 p. (1991).

A reflexive and symmetric relation \(\rho\subset A^ 2\) is a tolerance of the algebra \((A,\Omega)\) if \(\rho\) is compatible with the operations of \((A,\Omega)\). What can tolerances do for you? Transitive tolerances are comgruence relations which are an essential tool for decomposing algebras into direct and subdirect products and describing the equational theories. As tolerances generalize congruences we would expect a similar tool which provides us with more general but also weaker results.

This monograph goes far beyond these expectations. In the second chapter, classes and blocks of tolerances are studied and the theorem of Czedli is proved that for a tolerance \(T\) on a lattice \(L\) the quotient structure \(L/T\) is a lattice again. It follows a bunch of results on algebras and lattices with distributive or modular lattices of tolerances. In the next chapter conditions for algebras are pointed out for which every tolerance is a congruence. It is shown that this is the case whenever a variety is congruence permutable (Chajda). An algebra \(\underset\widetilde{} A\) is called tolerance simple if \(\underset\widetilde{} A\) has only trivial tolerances. The results connected with order polynomial completeness are also contained in this chapter.

The tolerance extension property and the regularity of tolerances are the topics of the following chapter. Further highlights are the representation of tolerance lattices, direct decomposability of tolerances, Chinese remainder theorem, reconstruction of the algebra from its tolerance lattices. Finally, one finds a bibliography which is ordered by years and gives insight into the development of the topic. This is an excellent contribution to a modern topic of lattice theory.

This monograph goes far beyond these expectations. In the second chapter, classes and blocks of tolerances are studied and the theorem of Czedli is proved that for a tolerance \(T\) on a lattice \(L\) the quotient structure \(L/T\) is a lattice again. It follows a bunch of results on algebras and lattices with distributive or modular lattices of tolerances. In the next chapter conditions for algebras are pointed out for which every tolerance is a congruence. It is shown that this is the case whenever a variety is congruence permutable (Chajda). An algebra \(\underset\widetilde{} A\) is called tolerance simple if \(\underset\widetilde{} A\) has only trivial tolerances. The results connected with order polynomial completeness are also contained in this chapter.

The tolerance extension property and the regularity of tolerances are the topics of the following chapter. Further highlights are the representation of tolerance lattices, direct decomposability of tolerances, Chinese remainder theorem, reconstruction of the algebra from its tolerance lattices. Finally, one finds a bibliography which is ordered by years and gives insight into the development of the topic. This is an excellent contribution to a modern topic of lattice theory.

Reviewer: D.Schweigert (Kaiserslautern)

### MSC:

08-02 | Research exposition (monographs, survey articles) pertaining to general algebraic systems |

08A30 | Subalgebras, congruence relations |

06B10 | Lattice ideals, congruence relations |