Tue, 17 September 2024, 1:25 pm MT
The height of a (finite) lattice L is the size of a maximum chain in L. Cameron, Solomon and Turull (1989) showed that the height of the subgroup lattice of the symmetric group S_n is given by \lceil 3n/2 \rceil - b(n), where b(n) is the number of 1s in the binary expansion of n. The analogue of S_n for semigroups is the full transformation semigroup T_n. Cameron, Gadoleau, Mitchell and Perese (2017) established an accurate asymptotic formula for the height of the subsemigroup lattice of T_n. But the subgroup lattice of a group G can be viewed from a different angle: it is (isomorphic to) the lattice of right (or left) congruences of G.(And one-sided congruences of a monoid are in a 1-1 correspondence with cyclic transformation representations of S.) In this talk I will introduce a general method which gives a lower bound for the height of the lattices of one- or two-sided congruences of an arbitrary semigroup, and under certain additional conditions gives the exact values. I will apply this theory to obtain the height for the lattices of right and left congruences of T_n, as well as for many other natural semigroups of transformations, partitions and matrices. This is joint work with Matthew Brookes, James East, Craig Miller and James Mitchell.
[video]