[Paper Review] Two representation theorems of three-valued structures by means of binary relations

Luisa Iturrioz|ArXiv.org|Oct 4, 2007

Advanced Algebra and Logic5 references3 citations

TL;DR

This paper presents two representation theorems for three-valued structures known as T-structures by embedding them into algebras of binary relations. Using Stone-type duality, it establishes that every T-structure is isomorphic to a relational T-structure, where operations like interior, closure, and negation are represented via symmetric relations and set-theoretic operations on subsets of a set E.

ABSTRACT

The results here presented are a continuation of the algebraic research line which attempts to find properties of multiple-valued systems based on a poset of two agents. The aim of this paper is to exhibit two relationships between some three-valued structures and binary relations. The established connections are so narrow that two representation theorems are obtained.

Motivation & Objective

• To establish a representation of three-valued structures (T-structures) using binary relations, providing a concrete algebraic realization.
• To extend the duality theory of distributive lattices to T-structures with three unary operators: C (complement), S₁ (interior), and S₂ (closure).
• To demonstrate that T-structures can be fully represented as relational algebras, preserving all operations and identities.
• To formalize the connection between T-structures and rough set theory via relational interpretations.
• To provide a constructive isomorphism between an abstract T-structure and a concrete relational model using a set E and a symmetric relation G.

Proposed method

• Construct a Stone-type representation using the spectrum of prime filters (or prime ideals) of the underlying distributive lattice.
• Define a map h: A → G ∩ (f(A) × E), where f(a) is the set of prime filters containing a, and G is a symmetric relation on E.
• Represent the unary operators S₁, S₂, and C via relational operations: S₁(a) = h(a) ∩ h(a)⁻¹, S₂(a) = h(a) ∪ h(a)⁻¹, and C(a) = complement in the power set.
• Prove that the map h preserves all operations: meet, join, S₁, S₂, and C, using properties of filters and the determination principle.
• Verify that the image of h forms a T-structure of relations isomorphic to the original algebra, satisfying all axioms (T1)–(T7).
• Use the fact that S₂(a) = f(a) ∪ g(f(a)) and S₁(a) = f(a) ∩ g(f(a)) where g is the involution associated with the symmetric relation G.

Experimental results

Research questions

• RQ1 Can every T-structure be represented as a relational algebra of binary relations preserving all operations?
• RQ2 How can the three unary operators S₁, S₂, and C in a T-structure be concretely interpreted using relations on a set?
• RQ3 What is the role of symmetric relations and their inverses in modeling interior and closure operators in three-valued logic?
• RQ4 To what extent do the axioms of T-structures (T1)–(T7) correspond to relational set operations in the representation?
• RQ5 Is there a Stone-type duality that extends to T-structures with non-functional completeness, particularly in the context of rough sets and intuitionistic logic?

Key findings

• Every T-structure is isomorphic to a relational T-structure, where elements are subsets of a set E equipped with a symmetric relation G.
• The representation map h preserves all operations: h(a ∧ b) = h(a) ∩ h(b), h(a ∨ b) = h(a) ∪ h(b), h(S₁a) = S₁h(a), h(S₂a) = S₂h(a), and h(Ca) = Ch(a).
• The operator S₂ is represented as the union of a set and its inverse under G, i.e., S₂(R) = R ∪ R⁻¹, while S₁(R) = R ∩ R⁻¹.
• The image of the representation is closed under the relational operations, forming a subalgebra isomorphic to the original T-structure.
• The representation is faithful and injective, relying on the determination principle (T6), which ensures that equality in the images implies equality in the original algebra.
• The relational model satisfies all axioms (T1)–(T7), including the key identity S₁a ∨ Ca = 1 and S₁a ∧ Ca = 0, which are preserved via relational complementation and intersection.