Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleLong chains of subsemigroups
PresenterJames Mitchell (University of St Andrews)
Author(s)Dr James Mitchell

The length of a semigroup S is defined to be the largest size of a chain of
subsemigroups of S. An exact formula for the length of the symmetric group on n
points was found by Cameron, Solomon and Turull; the length is roughly 3n/2. In
general, it follows by Lagrange's Theorem that the length of a group is at most
the logarithm of the group order. Semigroups refuse to be as well-behaved. The
only valid upper bound for the length of an arbitrary semigroup is its size. For
example, any zero-semigroup has length equal to its size. Even for less
degenerate and more natural examples of semigroups, the contrast to groups is
noticable. We will see that the length of the full transformation semigroup on n
points, the semigroup analogue to the symmetric group, is asymptotically at
least a constant multiple of its size.