Principal congruences on some lattice-ordered algebras.

*(English)*Zbl 0702.06011The authors solve the question in which algebras all congruences are principal (PC). They prove:

1) A distributive lattice L has property (PC) if L is finite and all subsets of J(L) \((=\) the poset of all nonzero join-irreducible elements of L) are convex (resp. \(\ell (J(L))\leq 1).\)

2) A Stone algebra L has property (PC) if L is finite and \(\ell (J(L))\leq 2.\)

3) A de Morgan algebra has property (PC) if L is finite and \(\ell (J(L))\leq 3.\)

They also show that the Heyting algebras behave quite differently - to ensure that all congruences are principal a chain condition is necessary and sufficient.

1) A distributive lattice L has property (PC) if L is finite and all subsets of J(L) \((=\) the poset of all nonzero join-irreducible elements of L) are convex (resp. \(\ell (J(L))\leq 1).\)

2) A Stone algebra L has property (PC) if L is finite and \(\ell (J(L))\leq 2.\)

3) A de Morgan algebra has property (PC) if L is finite and \(\ell (J(L))\leq 3.\)

They also show that the Heyting algebras behave quite differently - to ensure that all congruences are principal a chain condition is necessary and sufficient.

Reviewer: R.Firlová

##### MSC:

06F25 | Ordered rings, algebras, modules |

06B10 | Lattice ideals, congruence relations |

06D05 | Structure and representation theory of distributive lattices |

06D20 | Heyting algebras (lattice-theoretic aspects) |

06D30 | De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) |

##### Keywords:

principal congruences; distributive lattice; join-irreducible elements; Stone algebra; de Morgan algebra; Heyting algebras; chain condition
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\textit{T. Blyth} and \textit{J. Varlet}, Discrete Math. 81, No. 3, 323--329 (1990; Zbl 0702.06011)

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