OFFSET
0,5
COMMENTS
Let H_f denote the H-class in the semigroup of partial transformations containing f. Then H_f contains an idempotent iff the image of f is a transversal for the kernel of f.
Let H_f ~ H_g iff the image of f is contained in the image of g and the kernel of f is more coarse than the kernel of g. Then ~ is a partial order on the H-classes, hence a preorder (quasi-order) on the semigroup. The poset is isomorphic to the Segre product of the Boolean lattice of rank n and the partition lattice of [n+1].
REFERENCES
O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, Chapter 4.4 - 4.6.
LINKS
FORMULA
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 21, 18, 1;
1, 60, 150, 40, 1;
1, 155, 900, 650, 75, 1;
...
MAPLE
T:= (n, k)-> binomial(n, k)*Stirling2(n+1, k+1):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Jun 24 2023
MATHEMATICA
Table[Table[Binomial[n, k] StirlingS2[n + 1, k + 1], {k, 0, n}], {n, 0, 5}] // Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jun 24 2023
STATUS
approved