Factorizable Submonoids of the Symmetric Inverse Monoid
by
Janusz Konieczny
Mary Washington College, Fredericksburg, Virginia
Coauthors: Stephen Lipscomb (Mary Washington College)

A monoid M is called factorizable if for every element a of M, there are an idempotent e in M and a unit u in M such that a=eu.

Let I_n be the symmetric inverse monoid of degree n, that is, the monoid of all partial one-to-one transformations on the set X_n={1, 2, ... , n}. The group of units of I_n is the symmetric group S_n of all permutations on X_n. For any permutation group G (subgroup of S_n), we define the monoid M(G) induced by G by:

M(G)={a in In : a is a restriction of some g in G}.

For every permutation group G, M(G) is a factorizable inverse monoid. It is the largest (with respect to inclusion) factorizable submonoid of I_n that has G as its group of units.

We study factorizable inverse submonoids of I_n induced by subgroups of S_n. Let G be a subgroup of S_n and let M=M(G). We give formulas for the order of M for some classes of groups G and investigate conjugacy classes of M using a generalization of the class equation for finite groups to finite monoids. In particular, we characterize the groups G for which the set of singleton conjugacy classes of M is the union of Z(G) and {0}, where Z(G) is the center of G and 0 is the zero transformation.

Date received: November 7, 2001