Partial Orders and Idempotents of Monoids
https://doi.org/10.22405/2226-8383-2021-22-4-183-199
Abstract
Idempotents of the monoid play different roles in the formation of its properties. A set of idempotents is divided into three parts: incomparable with a unit, less and equal to a unit,
and more and equal to a unit. The idempotents of the first part are called primary and the idempotents comparable with a unit are called secondary. The properties of idempotents are
investigated in terms of partial orders and Green’s equivalences. In the article the main attention is given to finding connections among different classical and non-classical, stable and unstable
partial orders and roles which the idempotents play in that. In particular, as a result, the criterion of stability of Mitsch’s partial order is obtained. Different examples of ordered monoids
are shown in the context of the constructed theory of idempotents and partial orders.
About the Author
Vladislav Bronislavovich Poplavski
Saratov State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
References
1. Vagner V. V.1952, "Generalized groups," Dokl. Akad. Nauk SSSR , no. 84, pp. 1119—1122.
2. Vagner V. V.1956, "Representation of ordered semigroups," Math. Sbornik , vol. 38(80), no. 2, pp. 203—240.
3. Mitsch H. 1986, "A Natural Partial Order for Semigroups," Proc. Amer. Math. Soc., vol. 97, no. 3, pp. 384—388.
4. Hartwig R. 1980, "How to partially order regular elements," Math. Japonica, vol. 25, no. 1, pp. 1—13.
5. Nambooripad K. 1980, "The natural partial order on a regular semigroup," Proc. Edin. Math. Soc., vol. 23, pp. 249—260.
6. Mitsch H. 1994, "Semigroups and their natural order," Math. Slovaca , vol. 44, no. 4 , pp. 445—462.
7. Poplavski V. B. 2012, "On idempotents of algebra of Boolean matrices ," Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., vol. 12, no. 2, pp. 26—33.
8. Poplavski V. B. 2016, "Divisibility for idempotents of semigroup of Boolean matrices," Math. Mech. Saratov Univ., no. 18, pp. 57-60.
9. Poplavski V. B. 2017, "On partial orders for the set of Boolean matrices," Electronic Information Systems, no. 3 (14), pp. 105—113.
10. Klifford A. H., Preston G. B. 1972 The Algebraic Theory of Semigroups. MIR, Moscow, vol. 1. 288 p.
11. Lallement G. 1985. Semigroups and Combinatorial Applications. MIR, Moscow. 440 p.
12. Miller D. D., Clifford A. H. 1956. "Regular 𝒟−classes in semigroups," Trans. Amer. Math. Soc., vol. 82, pp. 1—15.
13. Shchekaturova O. O. , VA Yaroshevich V. A.2013. "On the properties of Boolean matrices," Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., vol. 13, no 4(2), рp. 137-142.
14. Poplavski V. B., Javkaev D. G. "On Inverse D-classes of a Semigroup of Boolean Matrices," Proc. XVIth Int. Conf."Algebra, Number Theory and Discreet Geometry." Tula, 2019, pp.112-
15. http://poivs.tsput.ru/conf/international/XVI/files/Conference2019M.pdf
16. Poplavski V. B., Javkaev D. G. 2019. "Calculation of Inverse D-classes of Boolean Matrices," Math. Mech. Saratov Univ., no. 21, pp. 50-52.
17. Klifford A. H., Preston G. B. 1972. The Algebraic Theory of Semigroups. MIR, Moscow, vol. 2. 422 p.
18. Higgins P.M. 1994. "The Mitsch order on a semigroup," Semigroup Forum, vol.49, pp. 261—266.
Review
For citations:
Poplavski V.B. Partial Orders and Idempotents of Monoids. Chebyshevskii Sbornik. 2021;22(4):183-199. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-4-183-199
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