On Rank Properties of Endomorphisms of Finite Circular Orders

ABSTRACT

A mapping α from X = {1,2,…,n} to X is orientation-preserving if the sequence (1α,2α,…,nα) is a cyclic permutation of a nondecreasing sequence (with respect to some total order on X). Orientation-preserving mappings can be thought of as preserving a circular order on X. Two partitions of X have the same type if they have identical sizes and numbers of classes. Let τ be a partition with r classes, and let S be the semigroup generated by the set of orientation-preserving mappings from X to X with kernels of same type as τ. We show that the minimum cardinality of a generating set for S is . Moreover, we characterize all such S generated by their idempotent elements (i.e., s ∈ S such that s ² = s), and show that the minimum number of idempotent elements required to generate S is .

Key Words:

• Circular order
• Minimum cardinality generating sets
• Minimum cardinality idempotent generating sets
• Orientation-preserving mapping

Mathematics Subject Classification:

• 20M20

Notes

Communicated by P. Higgins.