Nagata products of bimodules over residuated lattices

TL;DR

This paper explores the Nagata product construction for bimodules over residuated lattices, revealing how it can reconstruct bimodules and establishing an adjunction with certain posemigroups, thereby unifying various known constructions.

Contribution

It introduces a new perspective on the Nagata product for residuated bimodules and establishes an adjunction with posemigroups, unifying existing twist product frameworks.

Findings

Reconstruction of bimodules from Nagata products under certain conditions

Establishment of an adjunction between bimodules and posemigroups

Unification of various twist product constructions

Abstract

We study the (restricted) Nagata product construction, which produces a partially ordered semigroup from a bimodule consisting of a partially ordered semigroup acting on a (pointed) join semilattice. A canonical example of such a bimodule is given by a residuated lattice acting on itself by division, in which case the Nagata product coincides with the so-called twist product of the residuated lattice. We show that, given some further structure, a pointed bimodule can be reconstructed from its restricted Nagata product. This yields an adjunction between the category of cyclic pointed residuated bimodules and a certain category of posemigroups with additional structure, which subsumes various known adjunctions involving the twist product construction.