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**Modularity and distributivity of tolerance lattices of commutative inverse semigroups.**
*(English)*
Zbl 0581.20058

Since the problem of characterizing all semigroups in certain classes with a given type of congruence lattice has attracted wide attention providing numerous satisfying results, the same problem is now investigated for tolerance lattices. A tolerance on a semigroup S is a reflexive and symmetric subalgebra of \(S\times S\), i.e. a congruence relation on S which is not transitive. This paper studies necessary and sufficient conditions for a commutative inverse semigroup to have a modular or distributive lattice of tolerances. In a series of 16 Lemmas a characterization expressed mainly by order theoretical properties of idempotents is given in both cases. The corresponding results for congruence lattices of (arbitrary) Clifford semigroups were found by J. B. Fountain and P. Lockley [Semigroup Forum 14, 81-91 (1977; Zbl 0392.20041)]. As a consequence it is shown that for a semilattice S modularity and distributivity of the tolerance lattice of S are equivalent (as it is in the case of congruence lattices).

Reviewer: H.Mitsch

### MSC:

20M10 | General structure theory for semigroups |

20M14 | Commutative semigroups |

06B15 | Representation theory of lattices |

20M15 | Mappings of semigroups |

08A30 | Subalgebras, congruence relations |

### Keywords:

commutative inverse semigroup; modular or distributive lattice of tolerances; idempotents; congruence lattices; Clifford semigroups; semilattice### Citations:

Zbl 0392.20041
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\textit{B. Pondělíček}, Czech. Math. J. 35(110), 146--157 (1985; Zbl 0581.20058)

### References:

[1] | Chajda I.: Lattices of compatible relations. Arch. Math. (Brno) 13 (1977), 89-96. · Zbl 0372.08002 |

[2] | Chajda I., Zelinka B.: Lattices of tolerances. Čas. pěst. mat. 102 (1911), 10-24. ’ · Zbl 0354.08011 |

[3] | Pondělíček B.: Atomicity of tolerance lattices of commutative semigroups. Czech. Math. J. 33 (108) (1983), 485-498. · Zbl 0535.20041 |

[4] | Clifford A. H., Preston G. B.: The algebraic theory of semigroups. Amer. Math. Soc., Providence, R. I. Vol I(1961); Vol. II (1967). · Zbl 0111.03403 |

[5] | Zelinka B.: Tolerance in algebraic structures II. Czech. Math. J. 25 (100) (1975), 175-178. · Zbl 0316.08001 |

[6] | Ore O.: Structure and group theory II. Duke Math. J. 4 (1938), 247-269. · Zbl 0020.34801 |

[7] | Nieminen J.: Tolerance relations on join-semilattices. Glasnik Mat. (Zagreb), 12 (1977), 243-246. · Zbl 0376.06011 |

[8] | Dean R. A., Oehmke R. H.: Idempotent semigroups with distributive right congruence lattice. Pacific J. Math. 14 (1964), 1187-1209. · Zbl 0128.25003 |

[9] | Papert D.: Congruence relations in semilattices. J. London Math. Soc. 39 (1964), 723 - 729. · Zbl 0126.03802 |

[10] | Mitsch H.: Semigroups and their lattice of congruences. Semigroup Forum 26 (1983), 1 - 63. · Zbl 0513.20047 |

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