1 results
THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 28 June 2013, pp. 279-292
- Print publication:
- April 2014
-
- Article
-
- You have access
- Export citation
-
For a semigroup
$S$, let 
${S}^{1} $ be the semigroup obtained from 
$S$ by adding a new symbol 1 as its identity if 
$S$ has no identity; otherwise let 
${S}^{1} = S$. Mitsch defined the natural partial order 
$\leqslant $ on a semigroup 
$S$ as follows: for 
$a, b\in S$, 
$a\leqslant b$ if and only if 
$a= xb= by$ and 
$a= ay$ for some 
$x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.
