# Lax Invariant in Coalgebra

• Jie-lin Li
• Lei Fan
Conference paper
Part of the Advances in Soft Computing book series (AINSC, volume 54)

## Abstract

In [1], Bart Jacobs and Jasse Hughes have brought in a new kind of functor. They took the order on a functor as a new functor. Based on that, they defined and researched some new notions about bisimulation. We take this new functor into the research of invariant in coalgebra, get the definition of predicate invariant, then we define and research several new notions. In the last, we can reach some conclusion about invariant. It is worth pointing out that we find the sufficient condition to make two-way lax invariant and invariant coincide, and prove that the great lax invariant is exactly the largest fixed point of some special functor coalgebra in set category.

## Keywords

Invariant Lax predicate lifting Lax invariant Fixed point Final category

## References

1. 1.
Jacobs, B., Hughes, J.: Simulations in coalgebra. In: Gunm, H.P. (ed.) Coalgebraic Methods in Computer Science. Electronic Notes in Theoretical Computer Science, vol. 82(1), pp. 71–109. Elsevier, Amsterdam (2003)Google Scholar
2. 2.
Rutten, J.: Automata and coinduction:An exercise in coalgebra. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)
3. 3.
Worrell, J.: On coalgebras and final semantics, Ph.D.Thesis, Computing Laboratory, Oxford University (2000)Google Scholar
4. 4.
Worrell, J.: Toposes of coalgebras and hidden algebras. In: Jacobs, B., Moss, L., Reichel, H., Rutten, J. (eds.) Procedings of the CMCS 1998. Electronic Notes in Theoretical Computer Science, vol. 11 (1998)Google Scholar
5. 5.
Rutten, J.: Universal coalgebra: a theory of systems. Theoretical Computer Science 249, 3–80 (2000)
6. 6.
Worrell, J.: Toposes of coalgebras and hidden algebras. In: Jacobus, B., Moss, L., Reichel, H., Rutten, J. (eds.) Coalgebraic Methods in Computer Science, Amsterdam. Electronic Notes in Theoretical Computer Science, vol. 11 (1998)Google Scholar
7. 7.
Hughes, J.: A Study of Categories of Algebras and Coalgebras, Ph.D.Thesis, Camegie Mellon University (2001)Google Scholar
8. 8.
Jacobs, B.: Comprehension for coalgebras. In: Moss, L. (ed.) Coalgebraic Methods in Computer Science. Electronic Notes in Theoretical Computer Science, vol. 65(1). Elsevier, Amsterdam (2002)Google Scholar
9. 9.
Xiaocong, Z., Zhongmei, S.: A Survey on the Coalgebraic Methods in Computer Science. Journal of Software 14(10), 1661–1671 (2003)
10. 10.
Johnstone, P.T., Power, A.J., Tsujishita, T., Watanabe, H., Worrell, J.: On the structure of categories of coalgebras. Theoretical Computer Science 260, 87–117 (2001)
11. 11.
Jacobs, B.: Introduction to Coalgebra,Towards Mathematics of States and Observations, Draft (2005)Google Scholar
12. 12.
Xiaohui, L., Lei, F.: Weak Invariant and Restrict Product of LTS. Computer Engineering And Science 155(11), 134–136 (2007)Google Scholar
13. 13.
Lei, F.: The Study For Several Topics in Domain Theory,Ph.D.Thesis. Beijing Capital Normal University (2001)Google Scholar
14. 14.
Chongyou, Z., Lei, F., Hongbin, C.: Frame and Continuous Lattices. Capital Normal University Press, Beijing (2000)Google Scholar