Abstract
Residuated structures, bounded commutative residuated lattices in particular, play an important role in the study of algebraic structures of logics—classical and non-classical. In this paper, by introducing partial adjoint pairs, a new structure is presented, named partial residuated lattices, which can be regarded as a version of residuated lattices in the case of partial operations, and their basic properties are investigated. The relations between partial residuated lattices and certain quantum structures are considered. We show that lattice effect algebras and D-lattices both are partial residuated lattices. Conversely, under certain conditions partial residuated lattices are both lattice effect algebras and D-lattices. Finally, dropping the assumption on commutativity, some similar results are obtained.
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References
Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer/Ister Science, Dordrecht/Bratislava
Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. I. Basic properties. Int J Theor Phys 40: 685–701
Dvurečenskij A, Vetterlein T (2001) Pseudoeffect algebras. II. Group representation. Int J Theor Phys 40: 703–726
Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24: 1331–1352
Grätzer G, Schmidt E (1963) Characterization of congruence lattices of abstract algebras. Acta Sci Math (Szeged) 24: 34–59
Grätzer G, Wenzel G (1967) On the concept of congruence relation in partial algebras. Math Scand 20: 275–280
Gudder S (1972) Partial algebraic structures associated with orthomodular posets. Pacific J Math 41: 717–730
Gudder S, Pulmannová S (1997) Quotients of partial Abelian monoids. Algebra Univers 38: 395–421
Gudder S, Schelp R (1970) Coordinatization of orthocomplemented and orthomodular posets. Proc Amer Math Soc 25: 229–237
Hájek P (1998) Metamathematics of fuzzy logic. Kluwer, Dordrecht
Jipsen P, Tsinakis C (2002) A survey of residuated lattices. In: Martinez J(eds) Ordered algebraic structures.. Kluwer, Dordrecht
Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44: 21–34
Ma ZH, Wu JD, Lu SJ (2004) Pseudo-effect algebras and pseudo- difference posets. Int J Theor Phys 43: 1453–1460
Pulmannová S (1997) Congruences in partial Abelian semigroups. Algebra Univers 37: 119–140
Schelp R (1970) Partial Baer*-semigroups and partial Baer semigroups. Dissertation, Kansas State Univ. Manhattan, Kansas
Shang Y, Li YM, Chen MY (2004) Pseudo difference posets and pseudo Boolean D-posets. Int J Theor Phys 43: 2447–2460
Vetterlein T (2004) BL-algebras and quantum structures. Math Slovaca 54: 127–141
Vetterlein T (2005) BL-algebras and effect algebras. Soft Comput 9: 557–564
Wang GJ (2000) Theory of non-classic mathematical logic and approximate reasoning. Science in China Press, Beijing (in Chinese)
Wilce A (1995) Partial Abelian semigroups. Int J Theor Phys 34: 1807–1812
Wilce A (1998) Perspectivity and congruence in partial Abelian semigroups. Math Slovaca 48: 117–135
Zhou XN, Li QG, Wang GJ (2007) Residuated lattices and lattice effect algebras. Fuzzy Set Syst 158: 904–914
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Project supported by the NSF of China (No. 10771524).
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Zhou, X., Li, Q. Partial residuated structures and quantum structures. Soft Comput 12, 1219–1227 (2008). https://doi.org/10.1007/s00500-008-0283-2
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DOI: https://doi.org/10.1007/s00500-008-0283-2