SEMIGROUPS GENERATED BY PARTITIONS

Abstract

Let X be a nonempty set and X^2 be the Cartesian square of X.

Some semigroups of binary relations generated by partitions of X^2 are studied.

In particular, the algebraic structure of semigroups generated by the finest

partition of X^2 and, respectively, by the finest symmetric partition of X^2 are

described.

Keywords

• Partition of a set,semigroup of binary relations,band of semigroups

References

1. B. Albayrak, O. Givradze and G. Partenadze, Generating sets of the complete semigroups of binary relations defined by semilattices of the class Sigma_2(X; 4), Applied Mathematics, 9 (2018), 17-27.
2. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London- Amsterdam, 1976.
3. Z. Avaliani and Sh. Makharadze, Maximal subgroups of some classes of semigroups of binary relations, Georgian Math. J., 11(2) (2004), 203-208.
4. R. Chaudhuri and A. Mukherjea, Idempotent Boolean matrices, Semigroup Forum, 21 (1980), 273-282.
5. A. H. Clifford, A proof of the Montague-Plemmons-Schein theorem on maximal subgroups of the semigroup of binary relations, Semigroup Forum, 1 (1970), 272-275.
6. A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups. Part 1, Mathematical Surveys, No. 7, Amer. Math. Soc., Providence, R.I., 1961.
7. H. M. Devadze, Generating sets of the semigroup of all binary relations in a finite set, Dokl. Akad. Nauk BSSR, 12 (1968), 765-768.
8. Ya. I. Diasamidze, One-sided zeros of subsets of a semigroup of binary relations, Ukrainian Math. J., 42(5) (1990), 532-535.

9. Ya. I. Diasamidze, One-sided units of a subset of a semigroup of binary relations, Ukrainian Math. J., 42(8) (1990), 915-918.
10. Ya. I. Diasamidze, The structure of idempotent binary relations, Studies of semigroups (Russian), Leningrad. Gos. Ped. Inst., Leningrad, (1990), 25-28.
11. Ya. I. Diasamidze, Right units of complete semigroups of binary relations defined by complete X-semilattices generated by pairwise nonintersecting sets, Bull. Georgian Acad. Sci., 166(1) (2002), 23-26.
12. Ya. I. Diasamidze, Right units of complete semigroups of binary relations, defined by complete X-semilattices generated by chains, Bull. Georgian Acad. Sci., 167(2) (2003), 197-199.
13. Ya. Diasamidze and Sh. Makharadze, Complete Semigroups of Binary Relations, Kriter, Turkey, 2013.
14. Ya. I. Diasamidze, N. Aydin and A. Erdo˘gan, Generating set of the complete semigroups of binary relations, Applied Mathematics, 7 (2016), 98-107.
15. O. Givradze, Some properties of semigroup BX(D), defined by semilattice of 1(X; 4) class, Bull. Georgian Acad. Sci., 167(1) (2003), 43-46.
16. O. Givradze, Y. Diasamidze and N. Tsinaridze, Generated sets of the complete semigroup binary relations defined by semilattices of the finite chains, Trans. A. Razmadze Math. Inst., 172 (2018), 378-387.
17. P. Hell and J. Ne˘set˘ril, Graphs and Homomorphisms, Oxford Lecture Series in Mathematics and its Applications, Vol. 28, Oxford University Press, Oxford, 2004.
18. P. M. Higgins, Techniques of Semigroup Theory, with a foreword by G. B. Preston, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.
19. J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, 12, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
20. W. Imrich, S. Klavˇzar and D. F. Rall, Topics in Graph Theory: Graphs and Their Cartesian Product, A K Peters, Ltd., Wellesley, MA, 2008.
21. K. H. Kim and F. W. Roush, Inverses of Boolean matrices, Linear Algebra Appl., 22 (1978), 247-262.
22. F. Klein-Barmen, ¨ Uber eine weitere verallgemeinerung des verbandsbegriffes, Math. Z., 46 (1940), 472-480.
23. J. Konieczny, Reduced idempotents in the semigroup of Boolean matrices, J. Symbolic Comput., 20 (1995), 471-482.
24. J. Konieczny, A proof of Devadze’s theorem on generators of the semigroup of Boolean matrices, Semigroup Forum, 83(2) (2011), 281-288.
25. K. Kuratowski and A. Mostowski, Set Theory, with an Introduction to Descriptive Set Theory, Translated from the 1966 Polish original, Second, completely revised edition, Studies in Logic and the Foundations of Mathematics, Vol. 86, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWNPolish Scientific Publishers, Warsaw, 1976.
26. D. B. McAlister, Homomorphisms of semigroups of binary relations, Semigroup Forum, 3(2) (1971/72), 185-188.
27. R. McKenzie and B. M. Schein, Every semigroup is isomorphic to a transitive semigroup of binary relations, Trans. Amer. Math. Soc., 349(1) (1997), 271- 285.
28. J. S. Montague and R. J. Plemmons, Maximal subgroups of the semigroup of relations, J. Algebra, 13 (1969), 575-587.
29. C. Namnak and P. Preechasilp, Natural partial orders on the semigroup of binary relations, Thai J. Math., 4(3) (2006), 39-50.
30. O. Ore, Theory of equivalence relations, Duke Math. J., 9 (1942), 573-627.
31. R. J. Plemmons and B. M. Schein, Groups of binary relations, Semigroup Forum, 1 (1970), 267-271.
32. R. J. Plemmons and M. T. West, On the semigroup of binary relations, Pacific J. Math. 35 (1970), 743-753.
33. G. B. Preston, Any group is a maximal subgroup of the semigroup of binary relations on some set, Glasgow Math. J., 14 (1973), 21-24.
34. B. M. Schein, Regular elements of the semigroup of all binary relations, Semigroup Forum, 13(2) (1976/77), 95-102.
35. B. M. Shain, Representation of semigroups by means of binary relations, Mat. Sb. (N.S.), 60(102) (1963), 293-303.
36. T. Tamura, Operations on binary relations and their applications, Bull. Amer. Math. Soc., 70(1) (1964), 113-120.
37. P. M. Whitman, Lattices, equivalence relations and subgroups, Bull. Amer. Math. Soc., 52 (1946), 507-522.
38. K. A. Zaretskii, Regular elements of the semigroup of binary relations, Uspehi Mat. Nauk, 17 (1962), 177-179.
39. K. A. Zaretskii, The semigroup of binary relations, Mat. Sb. (N.S.), 61(103) (1963), 291-305.