# ON SOME CLASSES OF REGULAR ORDER SEMIGROUPS

Gao, Zhenlin;Zhang, Guijie

• Published : 2008.01.31
• 41 8

#### Abstract

Here, some classes of regular order semigroups are discussed. We shall consider that the problems of the existences of (multiplicative) inverse $^{\delta}po$-transversals for such classes of po-semigroups and obtain the following main results: (1) Giving the equivalent conditions of the existence of inverse $^{\delta}po$-transversals for regular order semigroups (2) showing the order orthodox semigroups with biggest inverses have necessarily a weakly multiplicative inverse $^{\delta}po$-transversal. (3) If the Green's relation $\cal{R}$ and $\cal{L}$ are strongly regular (see. sec.1), then any principally ordered regular semigroup (resp. ordered regular semigroup with biggest inverses) has necessarily a multiplicative inverse $^{\delta}po$-transversal. (4) Giving the structure theorem of principally ordered semigroups (resp. ordered regular semigroups with biggest inverses) on which $\cal{R}$ and $\cal{L}$ are strongly regular.

#### Keywords

regular order semigroup;inverse $^{\delta}po$-transversals;POR-semigroups;ORB-semigroups

#### References

1. T. S. Blyth and G. A. Pinto, On idempotent-generated subsemigroups of principally ordered regular semigroups, Semigroup Forum 65 (2003), 1-12 https://doi.org/10.1007/s00233-002-0018-0
2. T. S. Blyth and G. A. Pinto, Idempotents in principally ordered regular semigroups, Communications in Algebra 19 (1991), 1549-1563 https://doi.org/10.1080/00927879108824220
3. T. S. Blyth and G. A. Pinto, On ordered regular semigroup with biggest inverses, Semigroup Forum 54 (1997), 154-165 https://doi.org/10.1007/BF02676598
4. T. S. Blyth and R. McFadden, Regular semigroups with a multiplicative inverse transversal, Proc. Rog. Soc. Edinburgh 92A (1982), 253-270
5. J. M. Howie, An Introduction to Semigroups Theory, Academic Press. London, 1976
6. S. Tatsuhiko, Naturally ordered regular semigroups with maximum inverses, Roc. Edinburgh Math. Sec. 32 (1989), 33-39 https://doi.org/10.1017/S001309150000688X
7. Z. Gao, Naturally ordered abundant semigroups with adequate transversals, PU. M. A 14 (2003), no. 1-2, 35-50