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Nagata products of bimodules over residuated lattices

Adam Přenosil, Constantine Tsinakis

TL;DR

This work develops a module-theoretic approach to Nagata twist constructions for posemigroups and residuated lattices. It introduces restricted Nagata products, nuclei/conuclei-based recovery maps, and an adjunction between cyclic pointed residuated bimodules and structured posemigroups, then extends the framework to bilattices and sesquilattices via equivalences. The paper also connects twist products to twistable pairs of residuated lattices and shows how algebras of fractions for Boolean-pointed Brouwerian algebras arise as nucleus/conucleus images of restricted twist products. Overall, it provides a unified categorical perspective on twist, fraction, and bilattice constructions across two-sorted and one-sorted algebraic settings.

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.

Nagata products of bimodules over residuated lattices

TL;DR

This work develops a module-theoretic approach to Nagata twist constructions for posemigroups and residuated lattices. It introduces restricted Nagata products, nuclei/conuclei-based recovery maps, and an adjunction between cyclic pointed residuated bimodules and structured posemigroups, then extends the framework to bilattices and sesquilattices via equivalences. The paper also connects twist products to twistable pairs of residuated lattices and shows how algebras of fractions for Boolean-pointed Brouwerian algebras arise as nucleus/conucleus images of restricted twist products. Overall, it provides a unified categorical perspective on twist, fraction, and bilattice constructions across two-sorted and one-sorted algebraic settings.

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.
Paper Structure (9 sections, 22 theorems, 153 equations)

This paper contains 9 sections, 22 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Nuclei and conuclei on posemigroups
  3. Modules over posemigroups
  4. The Nagata product
  5. Recovering a pointed bimodule from its Nagata product
  6. The structural bimodule
  7. The Nagata equivalence for bilattices and sesquilattices
  8. Bimodules arising from residuated pairs
  9. Twist products and bimonoids of fractions

Key Result

Theorem 4.2

The Nagata product of a doubly residuated $\ell$-bimodule is a residuated $\ell$-semigroup with the lattice structure and the semigroup structure and the two residuals

Theorems & Definitions (67)

  • Definition 4.1: The Nagata product ${\mathbf{S} \mathbin{\bowtie} \mathbf{M}}$
  • Theorem 4.2: Residuated $\ell$-semigroups arising from bimodules
  • Definition 4.3: The restricted Nagata product ${\mathbf{S} \mathbin{\bowtie_{0}} \mathbf{M}}$
  • proof
  • Theorem 4.5: Residuated lattices arising from pointed bimodules
  • proof
  • Proposition 5.2: Recovering $\mathbf{S}$ inside ${\mathbf{S} \mathbin{\bowtie} \mathbf{M}}$
  • proof
  • Proposition 5.3: Embedding $\mathbf{S}$ into ${\mathbf{S} \mathbin{\bowtie} \mathbf{M}}$
  • proof
  • ...and 57 more