Representable Relation Algebras Research Articles

FIRST-ORDER AXIOMATISATIONS OF REPRESENTABLE RELATION ALGEBRAS NEED FORMULAS OF UNBOUNDED QUANTIFIER DEPTH

AbstractUsing 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.

The structure of generalized BI-algebras and weakening relation algebras

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras.

Non-representable relation algebras from vector spaces


 
 
 Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018). 
 
 

Various notions of represetability for cylindric and polyadic algebras

Abstract For β an ordinal, let PEAβ (SetPEAβ) denote the class of polyadic equality (set) algebras of dimension β. We show that for any infinite ordinal α, if is atomic, then for any n < ω, the n-neat reduct of , in symbols , is a completely representable PEAn (regardless of the representability of ). That is to say, for all non-zero , there is a and a homomorphism such that fa(a) ≠ 0 and for any for which exists. We give new proofs that various classes consisting solely of completely representable algebras of relations are not elementary; we further show that the class of completely representable relation algebras is not closed under ≡∞,ω. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CAns satisfying the Lyndon conditions coincides with the class of atomic algebras in ElScNrnCAω, where El denotes ‘elementary closure’ and Sc is the operation of forming complete subalgebras.

Varieties generated by completions

We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from a paper by R. Maddux in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in a joint paper by M. Frias and R. Maddux are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras.

NONREPRESENTABLE RELATION ALGEBRAS FROM GROUPS

AbstractA series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

Subcompletions of representable relation algebras

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.

On the consistency problem for modular lattices and related structures

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.

THERE IS NO FINITE-VARIABLE EQUATIONAL AXIOMATIZATION OF REPRESENTABLE RELATION ALGEBRAS OVER WEAKLY REPRESENTABLE RELATION ALGEBRAS

AbstractWe 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.

On the equational complexity of RRA

We prove that the equational complexity function for the variety of representable relation algebras is bounded below by a log-log function.

On canonicity and completions of weakly representable relation algebras

AbstractWe show that the variety of weakly representable relation algebras is neither canonical nor closed under Monk completions.

Weak representations of relation algebras and relational bases

AbstractIt 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.

Axiomatizability of positive algebras of binary relations

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 simple construction of representable relation algebras with non‐representable completions

AbstractWe give a simple new construction of representable relation algebras with non‐representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Weakly representable relation algebras form a variety

We prove that the class of weakly representable relation algebras is closed under homomorphic images, hence it is a variety. As a corollary we classify the subdirectly irreducible algebras in this class.

Weakly representable atom structures that are not strongly representable, with an application to first order logic

AbstractLetn > 2. A weakly representable relation algebra that is not strongly representable is constructed. It is proved that the set of all n by n basic matrices forms a cylindric basis that is also a weakly but not a strongly representable atom structure. This gives an example of a binary generated atomic representable cylindric algebra with no complete representation. An application to first order logic is given. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the variety question for wRRA

In this paper we consider a question of Jonsson [6] whether the class of weakly representable relation algebras is a variety. We prove that the class is closed under taking homomorphic images provided that a certain embedding condition obtains.

Canonical varieties with no canonical axiomatisation

We give a simple example of a varietyV\mathbf {V}of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the varietyRRA\mathbf {RRA}of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences. Using probabilistic methods of Erdős, we construct an infinite sequenceG0,G1,…G_0,G_1,\ldotsof finite graphs with arbitrarily large chromatic number, such that eachGnG_nis a bounded morphic image ofGn+1G_{n+1}and has no odd cycles of length at mostnn. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from theGnG_nsatisfies arbitrarily many axioms from a certain axiomatisation ofV (RRA)\mathbf {V}\ (\mathbf {RRA}), while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish thatV (RRA)\mathbf {V}\ (\mathbf {RRA})has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.

Squares in Fork Arrow Logic

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Nemeti, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.

Strongly representable atom structures of 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.