TITLE:
On the Monoid of All Order Preserving Full Contractions with a Fixed Set
AUTHORS:
Muhammad Mansur Zubairu, Zulyadaini D. Lawan
KEYWORDS:
Contraction Map, Order Preserving, Fixed Set, Rank
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.15 No.3,
July
4,
2025
ABSTRACT: In this paper, we consider the monoid
FixOCT(
[ n ],D
)
of all order-preserving full contraction mappings that fix a subset say
D
of a finite
n
-element chain
{
1,2,⋯,n }
. We characterize regularity, Green’s relations, and starred Green’s relations, and show that this monoid is left adequate. Furthermore, we determine the cardinality of
FixOCT(
[ n ],D
)
, the ranks of its two-sided ideals, and demonstrate that the ranks of the two-sided ideals and their corresponding Rees quotients are equal. Moreover, we deduce the rank of the monoid
FixOCT(
[ n ],D
)
.