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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:
[1] Adams, M.E., Principal congruences in De Morgan algebras, Proc. Edinburgh math. soc., 30, 415-421, (1987) · Zbl 0595.06013
[2] M.E. Adams and R. Beazer, The intersection of principal congruences on de Morgan algebras, preprint. · Zbl 0716.06006
[3] Balbes, R.; Dwinger, Ph., Distributive lattices, (1974), University of Missouri Press Columbia, Missouri · Zbl 0321.06012
[4] Davey, B.A.; Duffus, D., Exponentiation and duality, (), 43-96, NATO Advanced Study Institutes Series · Zbl 0509.06001
[5] Fraser, G.A.; Horn, A., Congruence relations in direct products, Proc. amer. math. soc., 26, 390-394, (1970) · Zbl 0241.08004
[6] Grätzer, G., General lattice theory, (1978), Birkhäuser Basel und Stuttgart · Zbl 0385.06015
[7] Lakser, H., Principal congruences of pseudocomplemented distributive lattices, Proc. amer. math. soc., 37, 32-36, (1973) · Zbl 0269.06005
[8] Priestley, H.A., The construction of spaces dual to pseudocomplemented distributive lattices, Quart. J. math. Oxford, 26, 215-228, (1975) · Zbl 0323.06013
[9] Priestley, H.A., Ordered sets and duality for distributive lattices, Ann. discrete math., 23, 39-60, (1984) · Zbl 0557.06007
[10] Sankappanavar, H.P., A characterization of principal congruences of De Morgan algebras and its applications, Math. logic in Latin America, (1980), North-Holland · Zbl 0426.06008
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