RESEARCH ARTICLE

Map composition generalized to coherent collections of maps

  • Herng Yi CHENG 1 ,
  • Kang Hao CHEONG , 1,2
Expand
  • 1. National University of Singapore High School of Mathematics and Science, Singapore 129957, Singapore
  • 2. Tanglin Secondary School, Singapore 127391, Singapore

Received date: 09 Oct 2013

Accepted date: 16 Sep 2014

Published date: 01 Apr 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Herng Yi CHENG , Kang Hao CHEONG . Map composition generalized to coherent collections of maps[J]. Frontiers of Mathematics in China, 2015 , 10(3) : 547 -565 . DOI: 10.1007/s11464-015-0435-5

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

DOI

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

Outlines

/