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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

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)
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References:
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