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Principal congruences on some lattice-ordered algebras. (English) Zbl 0702.06011
The 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.
Reviewer: R.Firlová

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)
Full Text: DOI
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