Geodesic
The cubical homology of trace monoids
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 108-122.

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This article contains an overview of the results of the author's study in the field of algebraic topology used in computer science. The relationship between the cubical homology groups of generalized tori and homology groups of partial trace monoid actions is described. Algorithms for computing the homology groups of asynchronous systems, Petri nets, and Mazurkiewicz trace languages are shown. The main results of the paper were reported at the International conference «Toric Topology and Automorphic Functions» (September, 5–10th, 2011, Khabarovsk, Russia). 
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A. A. Khusainov. The cubical homology of trace monoids. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 108-122. https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a9/

[1] H.-J. Baues, G. Wirsching, “Cohomology of small categories”, J. Pure Appl. Algebra, 38:2-3 (1985), 187–211 | DOI | MR | Zbl

[2] M.A. Bednarczyk M. A., Categories of Asynchronous Systems, Ph.D. Thesis, report 1/88, University of Sussex, 1988 $marek} http://www.ipipan.gda.pl/<nobr>$ \widetilde{ | Zbl

[3] V. Diekert, Y. Métivier,, “Partial Commutation and Traces”, Handbook of formal languages, 3, Springer-Verlag, New York, 1997, 457–533 | DOI | MR

[4] U. Fahrenberg,, “A Category of Higher-Dimensional Automata”, Foundations of software science and computational structures, Lecture Notes in Computer Science, 3441, Springer-Verlag, Berlin, 2005, 187–201 | DOI | MR | Zbl

[5] P. Gabriel, M. Zisman,, Calculus of fractions and homotopy theory, Springer-Verlag, Berlin, 1967 | MR | Zbl

[6] E. Goubault, The Geometry of Concurrency, Ph.D. Thesis, Ecole Normale Supérieure, 1995 $goubault} http://www.dmi.ens.fr/<nobr>$ \widetilde{ | Zbl

[7] E. Haucourt, “A Framework for Component Categories”, Electronic Notes in Theoretical Computer Science, 230 (2009), 39–69 http://www.elsevier.com/locate/entcs | DOI

[8] A. A. Husainov, “Homological dimension theory of small categories”, J. Math. Sci., 110:1 (2002), 2273-2321 | DOI | MR | Zbl

[9] A. A. Husainov, “On the homology of small categories and asynchronous transition systems”, Homology Homotopy Appl., 6:1 (2004), 439–471 http://www.rmi.acnet.ge/hha | DOI | MR | Zbl

[10] A. A. Husainov, “On the Leech dimension of a free partially commutative monoid”, Tbilisi Math. J., 1:1 (2008), 71–87 http://www.tcms.org.ge/Journals/TMJ/index.html | MR | Zbl

[11] A. A. Husainov, “The global dimension of a trace monoid ring”, Semigroup Forum, 82:2 (2011), 261–270 | DOI | MR | Zbl

[12] A. A. Husainov, The homology groups of a partial trace monoid action, arxiv:1111.0854 v1 [math.AT], Cornell Univ., New York, 2011, 30 pp. http://arxiv.org/abs/1203.3098v1

[13] A. A. Khusainov, “Homology groups of asynchronous systems, Petri nets, and trace languages”, Sib. Electron. Mat. Izv., 9 (2012), 13–44 http://mi.mathnet.ru/eng/semr341 | MR

[14] A. A. Khusainov, “Cubical homology and the Leech dimension of free partially commutative monoids”, Sb.: Math., 199:12 (2008), 1859–1884 | DOI | MR | Zbl

[15] A. A. Khusainov, “Homology groups of semicubical sets”, Sib. Math. J., 49:1 (2008), 180–190 | DOI | MR | Zbl

[16] A. A. Khusainov, V. E. Lopatkin, I. A. Treshchev, “Studying a mathematical model of parallel computation by algebraic topology methods”, J. Appl. Ind. Math., 3:3 (2009), 353–363 | DOI | MR

[17] J. Leech, “Cohomology theory for monoid congruences”, Houston J. Math., 11:2 (1985), 207 – 223 | MR | Zbl

[18] S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998 | MR

[19] A. Mazurkiewicz, “Trace theory”, Advances in Petri Nets 1986, Proceedings of an Advanced Course (Bad Honnef, 8.-19. September 1986), Lecture Notes in Computer Science, 255, Springer-Verlag, Berlin, 1987, 278–324 | DOI | MR

[20] B. Mitchell, “Rings with several objects”, Adv. Math., 8 (1972), 1-161 | DOI | MR | Zbl

[21] M. Nielsen, G. Winskel, “Petri nets and bisimulation”, Theoretical Computer Science, 153:1-2 (1996), 211–244 | DOI | MR | Zbl

[22] U. Oberst, “Homology of categories and exactness of direct limits”, Math. Z., 107 (1968), 87–115 | DOI | MR | Zbl

[23] L.Yu. Polyakova, “Resolutions for free partially commutative monoids”, Sib. Math. J., 48:6 (2007), 1038–1045 | DOI | MR | Zbl

[24] G. Winskel, M. Nielsen, “Models for Concurrency”, Handbook of Logic in Computer Science, 4, ed. Abramsky, Gabbay and Maibaum, Oxford University Press, 1995, 1–148 | MR