CAAL : Categorical Abstract Algebraic Logic : coordinatization is algebraization

• Autorzy: Voutsadakis G.
• Języki publikacji: EN

Abstrakty

EN

The methods of categorical abstract algebraic logic are employed to show that the classical process of the coordinatization of abstract (affine plane) geometry can be viewed under the light of the algebraization of logical systems. This link offers, on the one hand, a new perspective to the coordinatization of geometry and, on the other, enriches abstract algebraic logic by bringing under its wings a very well-known geometric process, not known hitherto to be related or amenable to its methods and techniques. The algebraization takes the form of a deductive equivalence between two institutions, one corresponding to affine plane geometry and the other to Hall ternary rings.

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2012
• Tom: Vol. 47
• Strony: 125--145
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Voutsadakis G.

• School of Mathematics and Computer Science, Lake Superior State University, Sault Sainte Marie, MI 49783, USA,

Bibliografia

• [1] M. Barr and C. Wells, Category Theory for Computing Science, Third Edition, Les Publications CRM, Montr´eal, 1999.
• [2] W.J. Blok and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, Vol. 77, No. 396, 1989.
• [3] L.M. Blumenthal, A Modern View of Geometry, Dover Publications, Inc., New York,1980.
• [4] F. Borceux, Handbook of Categorical Algebra, Vol. I, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1994.
• [5] F. Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amsterdam, 1995.
• [6] P.J. Cameron, Projective and Polar Spaces, Second Online Edition,http://www.maths.qmul.ac.uk/ pjc/pps/, September 2000.
• [7] J. Czelakowski, Protoalgebraic Logics, Trends in Logic-Studia Logica Library 10,Kluwer, Dordrecht, 2001.
• [8] J. Fiadeiro and A. Sernadas, Structuring Theories on Consequence, in Recent Trends in Data Type Specification, Donald Sannella and Andrzej Tarlecki, Eds., Lecture Notes in Computer Science, 332 (1988), pp. 44–72.
• [9] J.M. Font and R. Jansana, A General Algebraic Semantics for Sentential Logics,Lecture Notes in Logic, 332:7 (1996), Springer-Verlag, Berlin Heidelberg, 1996.
• [10] J.M. Font, R. Jansana, and D. Pigozzi, A Survey of Abstract Algebraic Logic, Studia Logica 74:1/2 (2003), pp. 13–97.
• [11] J.A. Goguen and R.M. Burstall, Introducing Institutions, in Proceedings of the Logic of Programming Workshop, E. Clarke and D. Kozen, Eds., Lecture Notes in Computer Science 164 (1984), pp. 221–256.
• [12] J.A. Goguen and R.M. Burstall, Institutions: Abstract Model Theory for Specification and Programming, Journal of the Association for Computing Machinery 39 (1992), pp. 95–146.
• [13] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.
• [14] W. Szmielew, From Affine to Euclidean Geometry An Axiomatic Approach, Kluwer, Dordtrecht, 1983.
• [15] G. Voutsadakis, Categorical Abstract Algebraic Logic: Equivalent Institutions, Studia Logica 74 (2003), pp. 275–311.
• [16] G. Voutsadakis, Categorical Abstract Algebraic Logic: Algebraizable Institutions,Applied Categorical Structures 10 (2002), pp. 531–568.