A natural partial order for semigroups

Mitsch, H.

doi:10.1090/S0002-9939-1986-0840614-0

A natural partial order for semigroups
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Proc. Amer. Math. Soc. 97 (1986), 384-388 Request permission

Abstract:

A partial order on a semigroup $(S, \cdot )$ is called natural if it is defined by means of the multiplication of $S$. It is shown that for any semigroup $(S, \cdot )$ the relation $a \leq b$ iff $a = xb = by$, $xa = a$ for some $x$, $y \in {S^1}$, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup $({T_X}, \circ )$ of all maps on the set $X$ are investigated.

References

T. S. Blyth and Gracinda M. S. Gomes, On the compatibility of the natural order on a regular semigroup, Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), no. 1-2, 79–84. MR 700501, DOI 10.1017/S0308210500016140
W. D. Burgess and R. Raphael, On Conrad’s partial order relation on semiprime rings and on semigroups, Semigroup Forum 16 (1978), no. 2, 133–140. MR 491395, DOI 10.1007/BF02194622
Gracinda M. S. Gomes, On left quasinormal orthodox semigroups, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 1-2, 59–71. MR 723097, DOI 10.1017/S0308210500015791
Robert E. Hartwig, How to partially order regular elements, Math. Japon. 25 (1980), no. 1, 1–13. MR 571255
J. B. Hickey, Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc. (2) 26 (1983), no. 3, 371–382. MR 722568, DOI 10.1017/S0013091500004442
J. M. Howie, An introduction to semigroup theory, L. M. S. Monographs, No. 7, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0466355
G. Kowol and H. Mitsch, Naturally ordered transformation semigroups, Monatsh. Math. 102 (1986), no. 2, 115–138. MR 861935, DOI 10.1007/BF01490204
Mark V. Lawson, The natural partial order on an abundant semigroup, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 2, 169–186. MR 892688, DOI 10.1017/S001309150002825X
D. B. McAlister, Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Austral. Math. Soc. Ser. A 31 (1981), no. 3, 325–336. MR 633441
K. S. Subramonian Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. (2) 23 (1980), no. 3, 249–260. MR 620922, DOI 10.1017/S0013091500003801
V. V. Vagner, Generalized groups, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 1119–1122 (Russian). MR 0048425