Locally Ordered Biordered Sets

摘要

It is shown that a representation introduced by Easdown and Hall [1010. Easdown , D. , Hall , T. E. ( 1984 ). Reconstructing some idempotent-generated semigroups from their biordered sets . Semigroup Forum. 29 : 207 - 216 . [CrossRef], [Web of Science ®]View all references] 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(E) 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.