Abstract
The map which takes an element of an ordered set to its principal ideal is a natural embedding of that ordered set into its powerset, a semilattice. If attention is restricted to all finite intersections of the principal ideals of the original ordered set, then an embedding into a much smaller semilattice is obtained. In this paper the question is answered of when this construction is, in a certain arrow-theoretic sense, minimal. Specifically, a characterisation is given, in terms of ideals and filters, of those ordered sets which admit a so-called minimal embedding into a semilattice. Similarly, a candidate maximal semilattice on an ordered set can be constructed from the principal filters of its elements. A characterisation of those ordered sets that extend to a maximal semilattice is given. Finally, the notion of a free semilattice on an ordered set is given, and it is shown that the candidate maximal semilattice in the embedding-theoretic sense is the free object.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Easdown, D. and Hall, T. E. (1984) Reconstructing some idempotent-generated semigroups from their biordered sets, Semigroup Forum 29, 207-216.
McElwee, B. N. (1998) Locally Ordered Bisets, Unpublished Doctoral Thesis, University of Sydney.
Nambooripad, K. S. S. (1979) Structure of Regular Semigroups. I, Mem. Amer. Math. Soc. 224.