The abstract model-theoretic concepts of compactness and Löwenheim-Skolem properties are investigated in the "softer" framework of pre-institutions [18]. Two compactness results are presented in this paper: a more informative reformulation of the compactness theorem for pre-institution transformations, and a theorem on natural equivalences with an abstract form of the first-order pre-institution. These results rely on notions of compact transformation, which are introduced as arrow-oriented generalizations of the classical, object-oriented notions of compactness. Furthermore, a notion of cardinal pre-institution is introduced, and a Löwenheim-Skolem preservation theorem for cardinal pre-institutions is presented.
@article{bwmeta1.element.bwnjournal-article-bcpv28z1p67bwm,
author = {Salibra, Antonino and Scollo, Giuseppe},
title = {Compactness and L\"owenheim-Skolem properties in categories of pre-institutions},
journal = {Banach Center Publications},
volume = {28},
year = {1993},
pages = {67-94},
zbl = {0792.03026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv28z1p67bwm}
}
Salibra, Antonino; Scollo, Giuseppe. Compactness and Löwenheim-Skolem properties in categories of pre-institutions. Banach Center Publications, Tome 28 (1993) pp. 67-94. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv28z1p67bwm/
[000] [1] H. Andréka, I. Németi and I. Sain, Abstract model theoretic approach to algebraic logic, manuscript, Mathematical Institute, Budapest, 1984.
[001] [2] E. Astesiano and M. Cerioli, Commuting between institutions via simulation, Univ. of Genova, Formal Methods Group, Technical Report no. 2, 1990.
[002] [3] G. Bernot and M. Bidoit, Proving the correctness of algebraically specified software: Modularity and Observability issues, in: M. Nivat, C. M. I. Rattray, T. Rus and G. Scollo (eds.), AMAST '91, Algebraic Methodology and Software Technology, Workshops in Computing, Springer, London 1992, 216-239.
[003] [4] C. C. Chang and H. J. Keisler, Model Theory, third ed., North-Holland, Amsterdam 1990.
[004] [5] H.-D. Ebbinghaus, Extended logics: the general framework, in: J. Barwise and S. Feferman (eds.), Model-Theoretic Logics, Springer, Berlin 1985, 25-76.
[005] [6] H.-D. Ebbinghaus, J. Flum and W. Thomas, Mathematical Logic, Springer, New York 1984. | Zbl 0556.03001
[006] [7] H. Ehrig, M. Baldamus, F. Cornelius and F. Orejas, Theory of algebraic module specification including behavioural semantics and constraints, in: M. Nivat, C. M. I. Rattray, T. Rus and G. Scollo (eds.), AMAST '91, Algebraic Methodology and Software Technology, Workshops in Computing, Springer, London 1992, 145-172.
[007] [8] A. J. M. van Gasteren, On the Shape of Mathematical Arguments, Lecture Notes in Comput. Sci. 445, Springer, Berlin 1990. | Zbl 0705.68004
[008] [9] J. A. Goguen and R. Burstall, Introducing institutions, in: E. Clarke and D. Kozen (eds.), Logics of Programs, Lecture Notes in Comput. Sci. 164, Springer, Berlin 1984, 221-256. | Zbl 0543.68021
[009] [10] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Math. 5, Springer, New York 1971.
[010] [11] J.-A. Makowski, Compactness, embeddings and definability, in: J. Barwise and S. Feferman (eds.), Model-Theoretic Logics, Springer, Berlin 1985, 645-716.
[011] [12] V. Manca and A. Salibra, On the power of equational logic: applications and extensions, in: H. Andréka, J. D. Monk and I. Németi (eds.), Algebraic Logic, Colloq. Math. Soc. János Bolyai 54, North-Holland, Amsterdam 1991, 393-412. | Zbl 0749.08004
[012] [13] V. Manca, A. Salibra and G. Scollo, Equational type logic, Theoret. Comput. Sci. 77 (1990), 131-159. | Zbl 0716.03022
[013] [14] V. Manca, A. Salibra and G. Scollo, On the expressiveness of equational type logic, in: C. M. I. Rattray and R. G. Clark (eds.), The Unified Computation Laboratory: Modelling, Specifications and Tools, Oxford Univ. Press, Oxford 1992, 85-100. | Zbl 0768.08003
[014] [15] J. Meseguer, General logics, in: H.-D. Ebbinghaus et al. (eds.), Logic Colloquium '87, North-Holland, Amsterdam 1989, 275-329. | Zbl 0691.03001
[015] [16] M. P. Nivela and F. Orejas, Initial behaviour semantics for algebraic specifications, in: D. T. Sannella and A. Tarlecki (eds.), Recent Trends in Data Type Specification, Lecture Notes in Comput. Sci. 332, Springer, Berlin 1988, 184-207. | Zbl 0679.68026
[016] [17] F. Orejas, M. P. Nivela and H. Ehrig, Semantical constructions for categories of behavioural specifications, in: H. Ehrig, H. Herrlich, H.-J. Kreowski and G. Preuß (eds.), Categorical Methods in Computer Science - with Aspects from Topology, Lecture Notes in Comput. Sci. 393, Springer, Berlin 1989, 220-245.
[017] [18] A. Salibra and G. Scollo, A soft stairway to institutions, in: M. Bidoit and C. Choppy (eds.), Recent Trends in Data Type Specification, Lecture Notes in Comput. Sci. 655, Springer, Berlin 1993, 310-329.
[018] [19] D. T. Sannella and A. Tarlecki, Specifications in an arbitrary institution, Inform. and Comput. 76 (1988), 165-210. | Zbl 0654.68017