Locally Ordered Biordered Sets

Abstract

It is shown that a representation φ introduced by Easdown and Hall [10] affords a natural construction of biordered sets that are locally like semilattices from those that are locally like ordered sets. If E is a biordered set which is locally like an ordered set, then the biordered set E(⟨Eφ⟩) is locally like a semilattice. In particular, the property of being locally like an ordered set is preserved. Various additional local finitary conditions are similarly preserved. The biordered sets of regular locally inverse semigroups are locally ordered, and Nambooripad's characterisation of these biordered sets is revisited. Characterisations of the biordered sets of normal bands are given, together with a description of normal bands.

Key Words:

• Biordered set
• Idempotents
• Locally inverse semigroup
• Locally like an ordered set
• Locally like a semilattice
• Normal band
• Partial algebra
• Semigroup

2000 Mathematics Subject Classification:

• 06A99
• 20M10
• 20M20

Notes

Communicated by D. Easdown.