A simple construction of representable relation algebras with non-representable completions

2009 ◽  
Vol 55 (3) ◽  
pp. 237-244
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Simple Construction ◽  
Relation Algebras ◽  
Representable Relation

Non-finite-axiomatizability results in algebraic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 832-843 ◽  
Author(s):  
Balázs Biró
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Equational Theory ◽  
Negative Answer ◽  
Relation Algebras ◽  
Representable Relation Algebra ◽  
Finite Set ◽  
Variable Symbol ◽  
Axiom Systems ◽  
Representable Relation

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.)

scholarly journals On representable relation algebras.

1964 ◽  
Vol 11 (3) ◽  
pp. 207-210 ◽  
Author(s):  
Donald Monk
Keyword(s):  
Relation Algebras ◽  
Representable Relation

Amalgamation in relation algebras

1998 ◽  
Vol 63 (2) ◽  
pp. 479-484 ◽  
Author(s):  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Equational Theory ◽  
Weak Form ◽  
Boolean Algebras ◽  
The Other ◽  
Relation Algebras ◽  
Boolean Algebras With Operators ◽  
Other Hand ◽  
Arrow Logic ◽  
Representable Relation

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.

Various notions of represetability for cylindric and polyadic algebras

2019 ◽  
Vol 56 (3) ◽  
pp. 335-363
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Relation Algebras ◽  
Polyadic Algebras ◽  
Algebras Of Relations ◽  
Neat Embeddings ◽  
Representable Relation ◽  
Elementary Closure

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.

scholarly journals Weak representations of relation algebras and relational bases

2011 ◽  
Vol 76 (3) ◽  
pp. 870-882
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
The Other ◽  
Relation Algebras ◽  
Representable Relation

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.

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

2016 ◽  
Vol 9 (3) ◽  
pp. 511-521 ◽  
Author(s):  
JEREMY F. ALM ◽  
ROBIN HIRSCH ◽  
ROGER D. MADDUX
Keyword(s):  
Relation Algebras ◽  
Equational Basis ◽  
Representable Relation

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.

Step by step – Building representations in algebraic logic

1997 ◽  
Vol 62 (1) ◽  
pp. 225-279 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Step Method ◽  
Finite Case ◽  
Representable Relation Algebra ◽  
Important Open Problem ◽  
Player Game ◽  
Representable Relation

AbstractWe 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.

scholarly journals On canonicity and completions of weakly representable relation algebras

2012 ◽  
Vol 77 (1) ◽  
pp. 245-262
Author(s):  
Ian Hodkinson ◽  
Szabolcs Mikulás
Keyword(s):  
Relation Algebras ◽  
Representable Relation

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

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