×

zbMATH — the first resource for mathematics

The Conrad program: from \(l\)-groups to algebras of logic. (English) Zbl 1337.06006
Lattice-ordered groups (\(l\)-groups for short) have a fundamental role in the study of algebras of logic. The term Conrad program refers to Paul Conrad’s approach to \(l\)-groups focusing on lattice theoretic properties of their lattices of convex \(l\)-subgroups.
In the paper under review the authors develop a Conrad type approach to the study of a large class of algebras of logic, showing that a substantial part of the Conrad program can be extended to residuated lattices that satisfy the identity \(x\setminus e\approx e/x\), called \(e\)-cyclic in the present paper. This variety encompasses several varieties of significance in algebraic logic. For any \(e\)-cyclic residuated lattice \(L\) let \(C(L)\) be the lattice of its convex subalgebras. A first result proved in this paper is that \(C(L)\) is an algebraic distributive lattice whose principal convex subalgebras form a sublattice.
A second result is that, in case \(L\) satisfies the left or right prelinearity law, a convex subalgebra \(H\) of \(L\) is prime iff the set of all convex subalgebras exceeding \(H\) is a chain under set-inclusion. Thus the lattice of principal convex subalgebras of \(L\) is a relatively normal lattice. It is also proved that a variety \(V\) of \(e\)-cyclic residuated lattices that satisfy either of the prelinearity laws is semilinear iff for every \(L\) in \(V\) all minimal prime convex subalgebras of \(L\) are normal.
In the final part of the paper the authors consider special classes of residuated lattices, e.g., residuated lattices in which convex subalgebras are normal, and GMV-algebras.

MSC:
06D35 MV-algebras
06F05 Ordered semigroups and monoids
06F15 Ordered groups
03G10 Logical aspects of lattices and related structures
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
08B15 Lattices of varieties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, M.; Conrad, P.; Martinez, J., The lattice of convex -subgroups of a lattice-ordered group, (Glass, A. M.W.; Holland, W. C., Lattice-Ordered Groups, (1989), D. Reidel Dordrecht), 105-127
[2] Anderson, M.; Feil, T., Lattice ordered groups, an introduction, (1988), D. Reidel Publishing Company · Zbl 0636.06008
[3] Bahls, P.; Cole, J.; Galatos, N.; Jipsen, P.; Tsinakis, C., Cancellative residuated lattices, Algebra Universalis, 50, 1, 83-106, (2003) · Zbl 1092.06012
[4] Balbes, R.; Dwinger, P., Distributive lattices, (1974), University of Missouri Press · Zbl 0321.06012
[5] Blount, K.; Tsinakis, C., The structure of residuated lattices, Internat. J. Algebra Comput., 13, 4, 437-461, (2003) · Zbl 1048.06010
[6] Conrad, P., The structure of a lattice-ordered group with a finite number of disjoint elements, Michigan Math. J., 7, 171-180, (1960) · Zbl 0103.01501
[7] Conrad, P., Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc., 99, 212-240, (1961) · Zbl 0099.25401
[8] Conrad, P., The lattice of all convex -subgroups of a lattice-ordered group, Czechoslovak Math. J., 15, 101-123, (1965) · Zbl 0135.06301
[9] Conrad, P., Lex-subgroups of lattice-ordered groups, Czechoslovak Math. J., 18, 86-103, (1968) · Zbl 0155.05902
[10] Conrad, P., Lattice ordered groups, an introduction, (1970), Tulane University Lecture Notes · Zbl 0213.31502
[11] Darnel, M. R., Theory of lattice-ordered groups, (1995), Marcel Dekker · Zbl 0810.06016
[12] Dvurečenskij, A., States on pseudo MV-algebras, Studia Logica, 68, 301-327, (2001) · Zbl 0999.06011
[13] Dvurečenskij, A., Pseudo MV-algebras are intervals in -groups, J. Aust. Math. Soc., 70, 427-445, (2002) · Zbl 1027.06014
[14] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated lattices: an algebraic glimpse at substructural logics, Stud. Logic Found. Math., (2007), Elsevier Amsterdam · Zbl 1171.03001
[15] Galatos, N.; Tsinakis, C., Generalized MV-algebras, J. Algebra, 283, 254-291, (2005) · Zbl 1063.06008
[16] Glass, A. M.W., Partially ordered groups, Ser. Algebra, (1999), World Scientific · Zbl 0933.06010
[17] Hart, J. B.; Tsinakis, C., Decompositions of relatively normal lattices, Trans. Amer. Math. Soc., 341, 519-548, (1994) · Zbl 0799.06019
[18] Jipsen, P.; Montagna, F., On the structure of generalized BL-algebras, Algebra Universalis, 55, 226-237, (2006) · Zbl 1109.06011
[19] Jipsen, P.; Tsinakis, C., A survey of residuated lattices, (Martinez, Jorge, Ordered Algebraic Structures, (2002), Kluwer Dordrecht), 19-56 · Zbl 1070.06005
[20] Kopytov, V. M., Lattice-ordered locally nilpotent groups, Algebra Logic, 14, 407-413, (1975)
[21] Kühr, J., Prime ideals and polars in DR-monoids and pseudo BL-algebras, Math. Slovaca, 53, 233-246, (2003) · Zbl 1058.06017
[22] Kühr, J., Representable dually residuated lattice-ordered monoids, Discuss. Math. Gen. Algebra Appl., 23, 115-123, (2003) · Zbl 1066.06008
[23] Kühr, J., Pseudo BL-algebras and drl-monoids, Math. Bohem., 128, 199-208, (2003) · Zbl 1024.06005
[24] Kühr, J., Ideals of noncommutative DR-monoids, Czechoslovak Math. J., 55, 97-111, (2005) · Zbl 1081.06017
[25] Kühr, J., Representable pseudo-BCK-algebras and integral residuated lattices, J. Algebra, 317, 354-364, (2007) · Zbl 1140.06008
[26] Martinez, J., Archimedean lattices, Algebra Universalis, 3, 247-260, (1973) · Zbl 0272.06013
[27] Martinez, J., Free products in varieties of lattice-ordered groups, Czechoslovak Math. J., 22, 97, 535-553, (1972) · Zbl 0247.06022
[28] Metcalfe, G.; Paoli, F.; Tsinakis, C., Ordered algebras and logic, (Hosni, H.; Montagna, F., Uncertainty and Rationality, Publ. Sc. Norm. Super. Pisa, vol. 10, (2010)), 1-85
[29] Montagna, F.; Tsinakis, C., Ordered groups with a conucleus, J. Pure Appl. Algebra, 214, 1, 71-88, (2010) · Zbl 1185.06012
[30] Monteiro, A., L’arithmetique des filtres et LES espaces topologiques, (Segundo Symposium de Matematics-Villavicencio, Mendoza, Buenos Aires, (1954)), 129-162 · Zbl 0058.38503
[31] Monteiro, A., L’arithmetique des filtres et LES espaces topologiques. I, II, Notas Logica Mat., 29-30, (1954) · Zbl 0058.38503
[32] Mundici, D., Interpretations of AFC*-algebras in łukasiewicz sentential calculus, J. Funct. Anal., 65, 15-63, (1986) · Zbl 0597.46059
[33] Reilly, N. R., Nilpotent, weakly abelian and Hamiltonian lattice ordered groups, Czechoslovak Math. J., 33, 348-353, (1983) · Zbl 0553.06020
[34] Snodgrass, J.; Tsinakis, C., The finite basis theorem for relatively normal lattices, Algebra Universalis, 33, 40-67, (1995) · Zbl 0819.06009
[35] Tsinakis, C., A unified treatment of certain aspects of the theories of lattice-ordered groups and semi prime rings via Brouwerian lattices and multiplicative lattices, (1979), University of California at Berkeley, Doctoral dissertation
[36] van Alten, C. J., Representable biresiduated lattices, J. Algebra, 247, 672-691, (2002) · Zbl 1001.06012
[37] Wille, A. M., Residuated structures with involution, (2006), Shaker Verlag Aachen · Zbl 1124.06011
[38] Wolfenstein, S., Valeurs normales dans ub groupe réticulé, Atti Accad. Naz. Lincei Rend., 44, 337-342, (1968) · Zbl 0174.06003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.