Results about crossed polysquares

Document Type : Research Paper

Author

Department of Mathematics, National University of Skills (NUS), Tehran, Iran

Abstract

Crossed polysquares are defined by Dehghanizadeh, Davvaz and Alp. Their properties and the generalization of results from intersecting squares to intersecting polysquares have been expressed and proved by them with the help of fundamental relations.
In the following, the concept of intersected polymodules and $\Gamma$-equivalent, intersected polymodules of polygroups are introduced and some properties are obtained from it. In addition, the concept of hypermultiplying fiber and crossed polysquares in the homotopy form of kernels has been studied. These results have extended the results related to intersecting squares to intersecting polysquares. In this article, homotopy crossed polysquares are studied as homotopies of cokernel, then the image of a crossed polymodule is considered and some results are proved that show the correspondence between crossed polymodules and crossed polysquares. In the continuation of the studies, we can check the results about the crossed 2-squares and then the crossed 2-polysquares. In addition, concepts about intersecting n-squares and intersecting n-polysquares can be expanded and studied.

Keywords

Main Subjects


[1] M. Alp, Actor of crossed modules of algebroids, Proceedings of the 16th Int. Conf. Jangjeon Math. soc., 16 (2005) 6–15.
[2] M. Alp, Pullback crossed modules of algebroids, Iran. J. Sci. Technol. Trans. A Sci., 32 no. 1 (2008) 1–5.
[3] M. Alp, Pullbacks of profinite crossed modules and CAT1-profinite groups, Algebras Groups Geom., 25 no. 2 (2008) 215–221.
[4] M. Alp and B. Davvaz, Crossed polymodules and fundamental relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 no. 2 (2015) 129–140.
[5] M. Alp and Ö. Gürmen, Pushouts of profinite crossed modules and cat 1-profinite groups, Turkish J. Math., 27 (2003) 539–548.
[6] Z. Arvasi and T. porter, Freeness conditions for 2-crossed modules of commutative algebras, Appl. Categ. Structures, 6 (1998) 455–471.
[7] Z. Arvasi and E. Ulualan, On algebraic models for homotopy 3-types, J. Homotopy Relat. Struct.,1 no. 1 (2006) 1–27.
[8] H. J. Baues, Combinatorial homotopy and 4-dimensional compexes, Walter de Gruyter, Berlin, De Gruyter expositions in Mathematics, 1991.
[9] R. Brown and G. H. Mosa, Double categories, R-categories and crossed modules, U. C. N. W. maths. preprint, 88 no. 11 (1988) 1–18.
[10] S. D. Comer, Extension of polygroups by polygroups and their representations using colour schemes, Lecture notes in Meth. 1004 (1982) 91–103.
[11] D. Conduché, Modules croisés généralisés de longueur 2, J. pure Applied Algebra, 34 (1984) 155–178.
[12] P. Corsini, Prolegomena of hypergroup theory, Supplement to Riv. Mat. Pura Appl., Aviani Editore, Tricesimo, 1993 215 pp.
[13] B. Davvaz, Isomorphism theorems of polygroups, Bull. Malays. Math. Sci. Soc. (2), 33 no. 3 (2010) 385-392.
[14] M. A. Dehghanizadeh, B. Davvaz and M. Alp, On crossed polysquares and fundamental relations, 13th Algebraic Hyperstructures and its Applications (AHA2017) 24-27 july, Istanbul-Turkey, (2017) and appeared in: Sigma J. Eng. & Nat. Sci., 9 no. 1 (2018) 1–16.
[15] M. A. Dehghanizadeh, B. Davvaz and M. Alp, On crossed polysquare version of homotopy kernels, Journal of Mathematical Extension, 16 no. 3 (2022) 1–37.
[16] D. Freni, Une note sur le coeur d’un hypergroupe et sur la clôture transitive β∗ de β, Riv. Math. Pura Appl., 8 (1991) 153–156.
[17] D. Guin-Walery and J. L. Loday, Obstruction á 1’excision en k-théories algébrique, In E. M. Friedlander, M. R. Stein (eds.), Evanston conf. On Algebraic k-Theory (1980), (Lect. Notes Math. 854) Springer, Berlin, Heidelberg, New York, (1981) 179–216.
[18] F. J. Korkes and J. Porter, Profinite crossed modules, U. C. N. W pure mathematics preprint, 86 no. 11 (1986).
[19] M. Koskas, Groupoids, demi-groups et hypergroups, J. Math. Pures Appl., 49 (1970) 155–192.
[20] V. Leoreanu-Fotea, The heart of some important classes of hypergroups, Pure math. Appl., 9 (1998) 351–360.
[21] J. L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Appl. Algebra, 24 (1982) 179–202.
[22] K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (1990) 129–146.
[23] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Inc, 115, palm Harber, USA, 1994.
[24] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc., 55 (1949) 453–496.