Representable distributive quasi relation algebras

TL;DR

This paper defines representability for distributive quasi relation algebras (DqRAs) via a poset-based construction that yields algebras as the lattice of up-sets $Up(oldsymbol{E})$ of a partially ordered equivalence relation. It establishes two equivalent representability notions (via equivalence and full DqRAs), proves that representable DqRAs coincide with isomorphic subalgebras of products of full DqRAs, and develops a concrete two-step construction using symmetry-enriched posets to produce both EDqRA and FDqRA. A variety of small finite representations are presented, including cases where nontrivial automorphisms enable finite representations, alongside nonrepresentable and finitely nonrepresentable examples and open questions about finite representability and the variety status. The work connects to weakening relation algebras, outlines generalizations to other FL-based algebras, and points to rich future directions in representability, structural theory, and related logical calculi.

Abstract

We give a definition of representability for distributive quasi relation algebras (DqRAs). These algebras are a generalisation of relation algebras and were first described by Galatos and Jipsen (2013). Our definition uses a construction that starts with a poset. The algebra is concretely constructed as the lattice of upsets of a partially ordered equivalence relation. The key to defining the three negation-like unary operations is to impose certain symmetry requirements on the partial order. Our definition of representable distributive quasi relation algebras is easily seen to be a generalisation of the definition of representable relations algebras by Jonsson and Tarski (1948). We give examples of representable DqRAs and give a necessary condition for an algebra to be finitely representable. We leave open the questions of whether every DqRA is representable, and also whether the class of representable DqRAs forms a variety. Moreover, our definition provides many other opportunities for investigations in the spirit of those carried out for representable relation algebras.

Figures (7)

• Figure 1: Two non-cyclic quasi relation algebras.
• Figure 2: A distributive lattice of binary relations from a poset.
• Figure 3: The twisted product from a three-element poset.
• Figure 4: Some representable chains up to $5$ elements. Here all chains are commutative and $0^3 = 0^2\cdot 0 = 0\cdot 0^2$. In (vi) and (vii) we have $0^3=0\cdot a=0^2\cdot a = 0^2$.
• Figure 5: Some $4$-element diamond representable quasi-relation algebras. All algebras listed here are commutative. In (i) we have $a^2 = 0 = 1=-1=1'$ and $-a = a' =a$. In (ii) we have $-a=b$, $-b = a$, $a'=a$ and $b'=b$, while in (iii) we have $-a = a'= b$ and $-b = b' = a$.

Theorems & Definitions (65)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof
• Proposition 2.6
• proof