On Ordered Regular Semigroups with a Zero Element

Document Type : Original Scientific Paper

Author

Department of Mathematics, College of Science, Sultan Qaboos University, Muscat, Oman

10.22052/mir.2022.243017.1301

Abstract

 In this paper, we study several conditions on ordered regular semigroups containing a zero element. In particular, we consider the natural and semigroup order and their connections to the properties of being principally ordered, Dubreil-Jacotin and BZS. We study also the set of biggest inverses in such a semigroup and we characterize subalgebras generated by two comparable idempotents.

Keywords


[1] T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
[2] T. S. Blyth and R. McFadden, Naturally ordered regular semigroups with a greatest idempotent,
Proc. Roy. Soc. Ed. Sect A 91 (1981) 107 - 122.
[3] T. S. Blyth and G. A. Pinto, Principally ordered regular semigroups,
Glasgow Math. J. 32 (1990) 389 - 416.
[4] T. S. Blyth and G. A. Pinto, Idempotents in principally ordered regular semigroups,
Comm. Algebra 19 (1991) 1549 - 1563.
[5] T. S. Blyth and G. A. Pinto, On ordered regular semigroups with biggest idempotents,
Semigroup Forum 54 (1997) 154 - 165.
[6] T. S. Blyth and G. A. Pinto, On idempotent-generated subsemigroups of principally ordered regular semigroups,
Semigroup Forum 68 (1) (2004) 47 - 58.
[7] T. S. Blyth and G. A. Pinto, Pointed principally ordered regular semigroups,
Discuss. Math. Gen. Algebra Appl. 36 (1) (2016) 101 - 111.
[8] P. A. Grillet,
Semigroups: An Introduction to the Structure Theory, Marcel Dekker, New York, 1995.
[9] J. M. Howie,
Fundamentals of Semigroup Theory, Oxford University Press Inc., New York, 1995.
[10] G. A. Pinto, Boolean zero square (BZS) semigroups,
SQU J. Sci. 26 (1) (2021) 31 - 39.
[11] T. Saito, Naturally ordered regular semigroups with maximum inverses,
Proc. Edinburgh Math. Soc. (2) 32 (1) (1998) 33 - 39.