Map composition generalized to coherent collections of maps

Abstract

Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set^∗, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements.

Keywords

Relation algebra / partial algebra / composition

References

[1] Belcastro S, Hull T C. Modelling the folding of paper into three dimensions using affine transformations. Linear Algebra Appl, 2002, 348: 273-282

[2] Börger R. Connectivity spaces and component categories. In: Categorical Topology, Proc Int Conf, Toledo/Ohio 1983. Sigma Ser Pure Math, 5. 1984, 71-89

[3] Cheng H Y, Cheong K H. Designing crease patterns for polyhedra by composing right frusta. Comput-Aided Design, 2012, 44: 331-342

[4] Crvenković S, Dolinka I. Varieties of involution semigroups and involution semirings: a survey. Bull Soc Math Banja Luka, 2002, 9: 7-47

[5] De Morgan A. On the syllogism: IV, and the logic of relations. T Cambridge Philos Soc, 1864, 10: 331-358

[6] Dolinka I. Idempotent distributive semirings with involution. Internat J Algebra Comput, 2003, 13: 597-625

[7] Dugowson S. On connectivity spaces. Cah Topol Géom Différ Catég, 2010, 51: 282-315

[8] Easdown D, Munn W D. On semigroups with involution. Bull Aust Math Soc, 1993, 48: 93-100

[9] Ésik Z, Bernátsky L. Equational properties of Kleene algebras of relations with conversion. Theoret Comput Sci, 1995, 137: 237-251

[10] Givant S. The calculus of relations as a foundation for mathematics. J Automat Reason, 2006, 37: 277-322

[11] Mac Lane S. Categories for the Working Mathematician. Berlin: Springer, 1978

[12] Maddux R D. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 1991, 50: 421-455

[13] Muscat J, Buhagiar D. Connective spaces. Mem Fac Sci Eng, Shimane Univ Ser B Math Sci, 2006, 39: 1-13

[14] Nummela E C. Cayley’s theorem for topological groups. Amer Math Monthly, 1980, 87: 202-203

[15] Reyes M L. Obstructing extensions of the functor spec to noncommutative rings. Israel J Math, 2012, 192: 667-698

[16] Tarski A. On the calculus of relations. J Symbolic Logic, 1941, 6: 73-89

[17] Taylor P. Practical Foundations of Mathematics. Cambridge: Cambridge University Press, 1999

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg