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**Geometrical methods in congruence modular algebras.**
*(English)*
Zbl 0547.08006

Mem. Am. Math. Soc. 286, 79 p. (1983).

Author’s abstract: We develop a geometric approach to algebras in congruence modular varieties. The idea of coordinatization of lines in affine geometry finds an almost perfect analog in the coordinatization of algebras. The geometry is the congruence class geometry, i.e. the subspaces are the blocks of congruence relations.

We show that congruence modularity guarantees that the congruence class geometry behaves nicely, because the Desarguesian and the Pappian theorems are true, if interpreted correctly. The innocuously looking ”Shifting Lemma” is the basic and powerful tool we need.

The obstacle to a perfect coordinatization is a congruence relation called the ”commutator”. The commutator is zero iff nonparallel lines have precisely one point of intersection.

This approach leads to a simple geometric development of commutator theory for arbitrary congruences. Results about affine algebras on the one hand and about distributive varieties on the other hand are tied together where only the commutator appears as a parameter. For the extreme values of this parameter we find theorems about affine, nilpotent and solvable congruences and varieties at one end and theorems generalizing Jónsson’s lemma at the other end. A radical, \(\sqrt{\underline A}\), is defined and we show that Jónsson’s lemma is true for every algebra \(\mathbf {A}/\sqrt{\underline A}\).

We show that congruence modularity guarantees that the congruence class geometry behaves nicely, because the Desarguesian and the Pappian theorems are true, if interpreted correctly. The innocuously looking ”Shifting Lemma” is the basic and powerful tool we need.

The obstacle to a perfect coordinatization is a congruence relation called the ”commutator”. The commutator is zero iff nonparallel lines have precisely one point of intersection.

This approach leads to a simple geometric development of commutator theory for arbitrary congruences. Results about affine algebras on the one hand and about distributive varieties on the other hand are tied together where only the commutator appears as a parameter. For the extreme values of this parameter we find theorems about affine, nilpotent and solvable congruences and varieties at one end and theorems generalizing Jónsson’s lemma at the other end. A radical, \(\sqrt{\underline A}\), is defined and we show that Jónsson’s lemma is true for every algebra \(\mathbf {A}/\sqrt{\underline A}\).

Reviewer: A. D. Bol’bot (Novosibirsk)

### MSC:

08B10 | Congruence modularity, congruence distributivity |

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

08B05 | Equational logic, Mal’tsev conditions |

08A30 | Subalgebras, congruence relations |

08A05 | Structure theory of algebraic structures |

51A25 | Algebraization in linear incidence geometry |

51A15 | Linear incidence geometric structures with parallelism |

51D15 | Abstract geometries with parallelism |