Abstract Barycentric Algebras

• Autorzy: Romanowska, A.B. , Smith, J.D.H. , Orłowska, E.
• Języki publikacji: EN

Abstrakty

EN

This paper presents a new approach to the study of (real) barycentric algebras, in particular convex subsets of real affine spaces. Barycentric algebras are cast in the setting of two-sorted algebras. The real unit interval indexing the set of basic operations of a barycentric algebra is replaced by an LP-algebra, the algebra of ukasiewicz Product Logic. This allows one to define barycentric algebras abstractly, independently of the choice of the unit real interval. It reveals an unexpected connection between barycentric algebras and (fuzzy) logic. The new class of abstract barycentric algebras incorporates barycentric algebras over any linearly ordered field, the B-sets of G. M. Bergman, and E. G. Manes' if-then-else algebras over Boolean algebras.

Słowa kluczowe

EN

affine space; barycentric algebra; mode; cancellavity mode; LII-algebra; Łukaszewicz Product Logic; heterogeneous algebra; if-then-else algebra; rectangular algebra

• Czasopismo: Fundamenta Informaticae
• Rocznik: 2007
• Tom: Vol. 81, nr 1-3
• Strony: 257-273
• Opis fizyczny: bibliogr. 32 poz.

Twórcy

autor

Romanowska, A.B.

autor

Smith, J.D.H.

autor

Orłowska, E.

• Warsaw University of Technology, 00-661 Warsaw, Poland,

Bibliografia

• [1] G. M. Bergman, Actions of Boolean rings on sets, Algebra Universalis 28 (1991), 153-187.
• [2] G. Birkhoff and J. D. Lipson, Heterogeneous algebras, J. Comb. Th. 8 (1970), 115-133.
• [3] W. J. Blok, I. M. A. Ferreirim, On the structure of hoops, Algebra Universalis 43 (2000), 233-257.
• [4] R. L. O. Cignoli, I. M. L. D'Ottaviano and D. Mundici, Algebraic Foundation of Many-valued Reasoning, Kluwer, Dordrecht, 2000.
• [5] P. Cintula, A note to the definition of ŁΠ-algebras, preprint (2004), to appear in Soft Computing.
• [6] F. Esteva, L. Godo and F. Montagna, The LΠ and LΠ1/2 logics: two complete fuzzy systems joining Łukasiewicz and product logics, Arch. Math. Logic 40 (2001), 39-67.
• [7] J. A. Goguen and J. Meseguer, Completeness of many-sorted equational logic, Houston J. Math. 11 (1985), 307-334.
• [8] S. P. Gudder, Convex structures and operational quantum mechanics, Comm. Math. Phys. 29 (1973), 249- 264.
• [9] P. Hájek, Metamathematics of Fuzzy Logic, Kluver, Dordrecht, 1998.
• [10] P. J. Higgins, Algebras with a scheme of operators, Math. Nachr. bf 27 (1963), 115-132.
• [11] V. V. Ignatov, Quasivarieties of convexors, (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 29 (1985), 12-14.
• [12] K. Isēki and S. Tanaka, An introduction to the theory of BCK-algebras, Mathematica Japonica 23, 1-26.
• [13] R. E. Jamison-Waldner, Functional representation of algebraic intervals, Pacific J. Math. 53 (1974), 399-423.
• [14] K. A. Kearnes, Idempotent simple algebras, in Logic and Algebra, Proc. of theMagariMemorial Conference, Siena, 1994, Dekker, New York, 1996, pp. 520-572.
• [15] K. Kearnes, Semilattice modes I: the associated semiring, Algebra Universalis 34 (1995), 220-272.
• [16] H. Lugowski, Grundzüge der Universallen Algebra, Teubner, Leipzig, 1976.
• [17] E. G. Manes, Adas and the equational theory of if-then-else, Algebra Universalis 30 (1993), 373-394.
• [18] J. A. McCarthy, A basis for a mathematical theory of computation, in (P. Braffort and D. Hirschberg, eds.), Computer Programming and Formal Systems, Northe-Holland, 1993, 33-70.
• [19] F.Montagna, An algebraic approach to propositional fuzzy logic, Journal of Logic, Language and Information 9 (2000), 91-124.
• [20] F. Montagna, Subreducts of MV-algebras with product and product residuation, Algebra Universalis 53 (2005), 109-137.
• [21] W. D. Neumann, On the quasivariety of convex subsets of affine spaces, Arch. Math. 21 (1970), 11-16.
• [22] F. Ostermann and J. Schmidt, Der baryzentrische Kalkül als axiomatische Grundlage der affinen Geometrie, J. Reine Angew.Math. 224 (1966), 44-57.
• [23] R. Pöschel, M. Reichel, Projection algebras and rectangular algebras, in General Algebra and Applications (eds. K. Denecke and H.-J. Vogel), Heldermann Verlag, Berlin, 1993, 180-194.
• [24] K. Pszczoła, A. Romanowska, J. D. H. Smith, Duality for some free modes, Discussiones Mathematicae, 23 (2003), 45-61.
• [25] A.B. Romanowska and J.D.H. Smith, Modal Theory, Heldermann, Berlin, 1985.
• [26] A.B. Romanowska and J.D.H. Smith, On the structure of barycentric algebras, Houston J.Math. 16 (1990), 431-448.
• [27] A.B. Romanowska and J.D.H. Smith, Modes, World Scientific, Singapore, 2002.
• [28] L.A. Skornyakov, Stochastic algebras, Izv. Vyssh. Uchebn. Zaved. Mat. 29 (1985), 3-11.
• [29] T.Stokes, Sets with B-action and linear algebra, Algebra Universalis 39 (1998), 31-43.
• [30] T. Stokes, Radical classes of algebras with B-action, Algebra Universalis 40 (1998), 73-85.
• [31] W. Wechler, Universal Algebra for Computer Scientists, Springer Verlag, Berlin, 1992.
• [32] A. Zamojska-Dzienio,Medial modes and rectangular algebras, Comment.Math. Univ. Carolinae 47 (2006), 21-34.