SAMS Subject Classification Number: 4

The category of (discrete) semilattices (i.e., idempotent commutative monoids) will be denoted by \(\mathord{\mathbf{SL}}\) and the category of compact \(0\)-dimensional topological semilattices will be denoted by \(\mathord{\mathbf{ZS}}\). The Duality Theorem for Semilattices (sometimes called “Pontryagin Duality for Semilattices,” see [1] and [2]) states that the hom-functors (suitably enriched) \[\mathord{\mathbf{SL}}(\underline{\phantom{x}}, 2):\mathord{\mathbf{SL}}\to \mathord{\mathbf{ZS}}\quad\text{and}\quad\mathord{\mathbf{ZS}}(\underline{\phantom{x}}, 2):\mathord{\mathbf{ZS}}\to \mathord{\mathbf{SL}}\qquad{(1)}\] provide a dual equivalence of categories. A straightforward proof from first principles is difficult to extract from the literature. In this talk, I sketch such a proof.

[1] Austin, C.W., 1963. Duality theorems for some commutative semigroups. Transactions of the American Mathematical Society, 109(2), pp.245-256.

[2] Hofmann, K.H., Mislove, M. and Stralka, A., 2006. The Pontryagin duality of compact 0-dimensional semilattices and its applications (Lecture Notes in Mathematics 396). Springer.