Monk algebras and representability

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

In 2007, Maddux observed that certain classes of representable relationalgebras (RRAs) form sound semantics for some relevant logics. In particular,(a) RRAs of transitive relations are sound for R, and (b) RRAs of transitive,dense relations are sound for RM. He asked whether they were complete aswell. Later that year I proved a modest positive result in a similar direction,namely that weakly associative relation algebras, a class (much) larger thanRRA, is sound and complete for positive relevant logic B. In 2008, Mikulas proved anegative result: that RRAs of transitive relations are not complete for R.His proof is indirect: he shows that the quasivariety of appropriatereducts of transitive RRAs is not finitely based. Later Maddux re-establishedthe result in a more direct way. In 2010, Maddux proved a contrasting positive result: that transitive, dense RRAs are complete for RM. He found an embedding of Sugihara algebras into transitive, dense RRAs. I will show that if we give up the requirement of representability, the positive result holds for R as well. To be precise, the following theorem holds.Theorem. Every normal subdirectly irreducible De Morgan monoid in the languagewithout Ackermann constant can be embedded into a square-increasing relationalgebra. Therefore, the variety of such algebras is sound and complete for R.

aRa is an automatic theorem prover for various kinds of relation algebras. It is based on Gordeev’s Reduction Predicate Calculi for n-variable logic (RPC n ) which allow first-order finite variable proofs. Employing results from Tarski/Givant and Maddux we can prove validity in the theories of simple semi-associative relation algebras, relation algebras and representable relation algebras using the calculi RPC3 , RPC4 and RPC ω . aRa, our implementation in Haskell, offers different reduction strategies for RPC n , and a set of simplifications preserving n-variable provability.

We prove the existence of non-representable relation algebras the union and complementation free reducts of which can be represented, i.e. which are weakly representable. This answers Problem 3 in Jonsson [4], and has consequences concerning the complexity of the equational theory of representable relation algebras.

A relation algebra atom structure α \alpha is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra C m α \mathfrak {Cm} \alpha is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982). Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ \Gamma , we construct a relation algebra atom structure α ( Γ ) \alpha (\Gamma ) and prove, for infinite Γ \Gamma , that α ( Γ ) \alpha (\Gamma ) is strongly representable if and only if the chromatic number of Γ \Gamma is infinite. A construction of Erdös shows that there are graphs Γ r \Gamma _r ( r > ω r>\omega ) with infinite chromatic number, with a non-principal ultraproduct ∏ D Γ r \prod _D\Gamma _r whose chromatic number is just two. It follows that α ( Γ r ) \alpha (\Gamma _r) is strongly representable (each r > ω r>\omega ) but ∏ D ( α ( Γ r ) ) \prod _D(\alpha (\Gamma _r)) is not.

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 (Algebra Univers 69:1–21, 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 Jónsson and Tarski (AMS 54:89, 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.

A finite axiom set for the identity-free equations valid in relation algebras is given. This is a simplification of the one given by Jonsson, and confirms a conjecture of Tarski. An axiom set for the identity-free equations valid in the representable relation algebras is given, too. We show that in the class of representable relation algebras, both the operation of taking converse and the identity constant are finitely axiomatizable (over the rest of the operations).

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.Other instances of this approach are looked at, and include the step by step method.

Many finite symmetric integral non-representable relation algebras, including almost all Monk algebras, can be embedded in the completion of an atomic symmetric integral representable relation algebra whose finitely-generated subalgebras are finite.

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.

We consider all positive fragments of Tarski’s representable relation algebras and determine whether the equational and quasiequational theories of these fragments are finitely axiomatizable in first-order logic. We also look at extending the signature with reflexive, transitive closure and the residuals of composition.

A survey of results is presented on relationships between the algebraic systems derived from the approximation spaces induced by information systems and various classes of algebras of relations. Rough relation algebras are presented and it is shown that they form a discriminator variety. A characterisation of the class of representable rough relation algebras is given. The family of closure operators derived from an approximation space is abstractly characterised as certain type of Boolean algebra with operators. A representation theorem is given which says that every such an algebra is isomorphic with a similar algebra that is derived from an information system.

The consistency problem for a class of algebraic structures asks for an algorithm to decide, for any given conjunction of equations, whether it admits a non-trivial satisfying assignment within some member of the class. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 533–535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172–194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839–874]. We derive unsolvability of the consistency problem for the class of (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite-dimensional vector spaces over a fixed or arbitrary field of characteristic [Formula: see text] and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann–Cayley algebra and to functional and embedded multivalued dependencies in databases.