Abstract
A residuated ordered algebra is a partially ordered set with additional ‘residuated’ operations. A construction is presented that, from any partial subalgebra of a residuated ordered algebra, constructs a complete algebra into which the partial subalgebra embeds. Conditions are given under which the constructed algebra is finite whenever a finite partial subalgebra is chosen. This implies the ‘finite embeddability property’ for the given class of residuated ordered algebras. In the case that the whole algebra is chosen as the partial subalgebra, the construction is a completion of the underlying order of the algebra. A scheme of inequalities is described that are shown to have the property of being preserved by the above construction. These preservation results thus extend the results on the finite embeddability property and completion.
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References
Blok W.J., van Alten C.J.: The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Algebra Universalis 48(3), 253–271 (2002)
Blok W.J., van Alten C.J.: On the finite embeddability property for residuated ordered groupoids. Trans. Amer. Math. Soc. 357(10), 4141–4157 (2005)
Blyth T.S., Janowitz M.F.: Residuation Theory. Pergamon Press, Oxford-New York (1972)
Ciabattoni, A., Galatos, N., Terui, K. From axioms to analytic rules in nonclassical logics. Proceedings of LICS’08, 229–240 (2008)
Davey B.A., Priestly H.A.: Introduction to Lattices and Order, 2nd edition. Cambridge University Press, Cambridge (2002)
Dilworth R.P.: Non-commutative residuated lattices. Trans. Amer. Math. Soc. 46, 426–444 (1939)
Evans T.: Some connections between residual finiteness, finite embeddability and the word problem. J. London Math. Soc. 1, 399–403 (1969)
Fuchs, L.: Partially Ordered Algebraic Systems. Addison-Wesley (1963)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, Volume 151, Elsevier (2007)
Galatos, N., Jipsen, P.: Residuated frames with applications to decidability. Trans. Amer. Math. Soc., to appear
Galatos, N., Ono, H.: Cut elimination and strong separation for substructural logics: an algebraic approach. Annals of Pure and Applied Logic, to appear
Gehrke M., Harding J., Venema Y.: MacNeille completions and canonical extensions. Trans. Amer. Math. Soc. 358(2), 573–590 (2006)
Givant S., Venema Y.: The preservation of Sahlqvist equations in completions of boolean algebras with operators. Algebra Universalis 41, 47–84 (1999)
Higman G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2(3), 326–336 (1952)
Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Ordered algebraic structures, 19–56, Dev. Math., 7, Kluwer Acad. Publ., Dordrecht (2002)
Jonsson B.: On the canonicity of Sahlqvist identities. Studia Logica 53, 473–491 (1994)
Kowalski, T., Ono, H.: Fuzzy logics from substructural perspective. In: Proceedings of the 26th Linz Seminar on Fuzzy Set Theory (2006)
MacNeille H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42(3), 416–460 (1937)
Terui K.: Which structural rules admit cut elimination? An algebraic criterion. J. Symbolic Logic 72(3), 738–754 (2007)
van Alten C.J.: The finite model property for knotted extensions of propositional linear logic. J. Symbolic Logic 70(1), 84–98 (2005)
van Alten C.J.: Axiomatizations of subreducts of n-potent residuated lattices. Algebra Universalis 57(1), 47–62 (2007)
van Alten C.J., Raftery J.G.: Rule separation and embedding theorems for logics without weakening. Studia Logica 76, 241–274 (2004)
Ward M., Dilworth R.P.: Residuated lattices. Trans. Amer. Math. Soc. 45, 335–354 (1939)
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Presented by I. Hodkinson.
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van Alten, C.J. Completion and finite embeddability property for residuated ordered algebras.. Algebra Univers. 62, 419–451 (2009). https://doi.org/10.1007/s00012-010-0060-9
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DOI: https://doi.org/10.1007/s00012-010-0060-9