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A new perspective on the tensor product of semi-lattices

Eric Buffenoir

Abstract

We adopt a new perspective on the tensor product of arbitrary semi-lattices. Our basic construction exploits a description of semi-lattices in terms of bi-extensional Chu spaces associated to a target space defined to be the boolean domain. The comparison between our tensor product and the canonical tensor product, introduced by G.A. Fraser, is made in the distributive case and in the general case. Some properties of our tensor products are also given.

A new perspective on the tensor product of semi-lattices

Abstract

We adopt a new perspective on the tensor product of arbitrary semi-lattices. Our basic construction exploits a description of semi-lattices in terms of bi-extensional Chu spaces associated to a target space defined to be the boolean domain. The comparison between our tensor product and the canonical tensor product, introduced by G.A. Fraser, is made in the distributive case and in the general case. Some properties of our tensor products are also given.
Paper Structure (15 sections, 42 theorems, 127 equations)

This paper contains 15 sections, 42 theorems, 127 equations.

Table of Contents

  1. Inf semi-lattices and States/Effects Chu spaces
  2. States/Effects Chu spaces
  3. Particular spaces of states
  4. Reduction of the space of effects
  5. Morphisms
  6. A new perspective on the construction of the tensor product of semi-lattices
  7. The canonical tensor product construction
  8. The maximal tensor product
  9. The minimal tensor product
  10. Canonical vs. minimal tensor product
  11. Properties of the minimal tensor product
  12. From maximal to regular tensor product
  13. Regular vs. minimal tensor product
  14. Channels of the bipartite experiments
  15. Conclusion

Key Result

Theorem 1.2

If the space of effects ${ {E}}$ satisfies the conditions of Definition DefinreductionE, then $({ {S}}, { {E}}, \epsilon^{ {S}})$ is a well-defined States/Effects Chu space.

Theorems & Definitions (96)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Theorem 1.5
  • proof
  • Remark 1.6
  • ...and 86 more