Search

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables. [ABSTRACT FROM AUTHOR]

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, x can access y if an electromagnetic or (slower than light) mechanical signal could be sent from x to y. We use Prior's tense language of F and P representing causal accessibility and its converse relation. We consider two versions, one where the accessibility relation is reflexive and one where it is irreflexive. In either case it has been an open problem, for decades, whether the logic is decidable or axiomatisable. We make a small step forward by proving, in each case, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable and that the complexity of each problem is PSPACE-complete. A consequence is that the temporal logic of intervals with real endpoints under either the containment relation or the strict containment relation is PSPACE-complete, the same is true if the interval accessibility relation is "each endpoint is not earlier", or its irreflexive restriction. We provide a temporal formula that distinguishes between three-dimensional and two-dimensional Minkowski spacetime and another temporal formula that distinguishes the two-dimensional case where the underlying field is the real numbers from the case where instead we use the rational numbers. [ABSTRACT FROM AUTHOR]

We prove that any equational basis that defines representable relation algebras (RRA) over weakly representable relation algebras (wRRA) must contain infinitely many variables. The proof uses a construction of arbitrarily large finite weakly representable but not representable relation algebras whose “small” subalgebras are representable. [ABSTRACT FROM AUTHOR]

Hirsch and Hodkinson proved, for 3 ≤ m < ω and any k < ω. that the class SNrmCAm+k+1 is strictly contained in SNrmCAm+k and if A k ≥ 1 then the former class cannot be defined by any finite set of first-order formulas, within the latter class. We generalize this result to the following algebras of m-ary relations for which the neat reduct operator ... is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalize this result to allow the case where m is an infinite ordinal. using quasipolyadic algebras in place of polyadic algebras (with or without equality). [ABSTRACT FROM AUTHOR]

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions. [ABSTRACT FROM AUTHOR]

It is known that for all finite n ≤ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RAn and wRRA contains the other. [ABSTRACT FROM AUTHOR]

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations. [ABSTRACT FROM AUTHOR]

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ. we construct an n-dimensional atom structure 퓒(Γ), and prove, for infinite Γ. that 퓒(Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdös shows that there are graphs Γk (k < ω) with infinite chromatic number, but having a non-principal ultraproduct ΠD Γk whose chromatic number is just two. It follows that 퓒(Γk) is strongly representable (each k < ω) but ΠD 퓒 (Γk) is not. [ABSTRACT FROM AUTHOR]

We show, for any ordinal γ ≥ 3, that the class RaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games. Fn (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra A with countably many atoms, that ∃ has a winning strategy in Fω (At(A)) ⇔ A ∈ ScRaCAω, ∃ has a winning strategy in Fn (At(A)) ⇐ A ∈ ScRaCAn, ∃ has a winning strategy in G (At(At)) ⇐ A ∈ RaCAω, ∃ has a winning strategy in H (At(At)) ⇒ A ∈ RaRCAω, for 3 ≤ n < ω. We use these games to show, for γ ≥ 5 and any class K of relation algebras satisfying RaRCAγ ⊆ K ⊆ ScRaCA5, that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion RaCAγ ⊂ ScRaCAγ is strict. For infinite γ and for a countable relation algebra A we show that A has a complete representation if and only if A is atomic and ∃ has a winning strategy in F(At(A)) if and only if A is atomic and A ∈ ScRaCAγ. [ABSTRACT FROM AUTHOR]