Maddux, R. (1982). Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272:501--526.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....RA: in fact, it is known that none of the axioms (R1) R4) R6) and (R7) holds in Ra CA 3 . Chapter 4 gives a complete description of those subsets of the RA axioms that can fail in the RA reduct of a CA 3 . Motivated by the failure of the associative law (R1) in Ra CA 3 , Maddux [Mad78] [Mad82] introduced the weaker axiom (R1 0 ) x ; 1) 1 = x ; 1, called the semiassociative law. Roughly speaking, R1 0 ) is how much of the associative law (R1) can be proved in CA 3 . Maddux defined the class SA of semiassociative relation algebras as the class of RA type algebras which satisfy (R1 ....

....4.3, 4.4(i) for condition (i) Theorem 4.6 (for condition (ii) and Theorem 4.4(ii) for condition (iii) Now suppose that Ra CA 3 ModE = E. Then cannot be (R1) since (R1) is not implied by (R4) R6) R7) in SRa CA 3 (this is because SA RA, proved in Maddux [Mad78] [Mad82]) If is (R4) then E must contain either (R6) or (R7) by Theorem 4.9. If is (R6) then E must contain (R7) by Theorem 4.8 and (R1) by Theorems 4.3, 4.7. Finally, if is (R7) then E must contain (R6) by Theorems 4.6, 4.8, and E must contain (R1) by Theorem 4.10. COROLLARY 4.12. A subset of ....

R.D. Maddux. Some varieties containing relation algebras. Trans. Amer. Math. Soc., 272:501--526, 1982.

....(see [Lyn56] it is valid for nite relation algebras. These so called Lyndon conditions can be seen as approximations to RRA. The Lyndon conditions can be expressed naturally in terms of a two player game. This was done in [HH97d, HH97b] but much the same ideas were used by Maddux, in [Mad82] for example, and indeed in [Lyn50] We believe that the explicit use of games concentrates attention on the essential concepts rather than notational details, and so permits more lucid proofs. The game is played on networks . An (atomic) network is a nite complete directed graph, with each ....

....indeed any atomic non associative algebra [Mad89] but we leave generalisations to the main part of the paper. Also, this is not Maddux s de nition, but it is equivalent: see proposition 31 below. consider are like those given by Maddux as semantics for the weakly associative relation algebras [Mad82] The unit (the interpretation of the top element 1 in the representation) is a re exive, symmetric relation on the base set M of the representation, and all operations, in particular the composition ; are relativised to the unit. So a pair (x; y) of elements of M will stand in the composite ....

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R D Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501{ 526, 1982.

....and x is the complement of x with respect to the universe. For expressions of relation algebras written without parenthesis, unary operators are executed first, followed by the composition, intersection, and union. Varieties on the associative property of relation algebras have been studied [1, 31]. They define semiassociative (Equation 3a) weakly associative (Equation 3b) and nonassociative (Equation 3c) algebras. Relation algebras (RA) semiassociative algebras (SA) weakly associative algebras(WA) and nonassociative algebras (NA) are related by Equation 3d. Most theorems defined for a ....

R. Maddux, "Some Varieties Containing Relation Algebras," Transactions of the American Mathematical Society, vol. 272, no. 2, pp. 501-526, 1982.

....been chosen by 9. This will be crucial later. THEOREM 1. Let A be a finite relation algebra. A is representable if and only if 9 has a winning strategy in G(A) Proof. See, for example, HH97b, theorem 9] or [HH97a, proposition 13] The idea is essentially in [Lyn50] and is well known (e.g. [Ma82]) 3. The Tiling Problem An instance of the tiling problem is a finite set of square tiles = fT 0 ; T k Gamma1 g. Each tile has a colour on each of its four edges: the four colours on the tile T i are Top(T i ) Bot(T i ) Lt(T i ) and Rt(T i ) See figure 1. Note that the tiles have a ....

....= e 0 e 1 e 2 follows from this definition of consistency. Clearly, we can obtain RA( from effectively (by an algorithm) LEMMA 2. For any instance of the tiling problem , RA( is a weakly associative algebra. Proof. Let C be the set of consistent triangles of RA( By theorem 2.2. 3 of [Ma82] 5 it suffices to show that (i) C is closed under Peircean transforms, ii) if (e i ; x; y) 2 C then x = y and (iii) for any atom a ij of RA( e i ; a ij ; a ji ) 2 C. This is rather easy to verify from the definition of the atom structure of RA( bearing in mind that the only inconsistent ....

R Maddux. Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272(2):501--526, 1982.

....algebras is not elementary. 1 Introduction There are two types of representation in algebraic logic: ordinary and complete representations. Ordinary representations, or just representations, have been studied extensively [J onsson and Tarski1948, Lyndon1950, Lyndon1956, McKenzie1970, Maddux1978, Maddux1982, Henkin et al..1985, Andr eka et al..1991, Venema1992, Monk1993] and are isomorphisms from a boolean algebra with operators to a more concrete structure where the boolean operators and Gamma are replaced by [ and n, and the other operators have certain set theoretically definable ....

R Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501--526, 1982.

....## E # (R1) R6) or E # (R1) R7) Proof. By theorem 2.2 ii and (2.2) R1) R6) and hence by theorem 2.2 iii (R1) R7) define RA in Ra # CA 3 . IfEdefines RA in Ra # CA 3 then (R1) # E because RA # SA (see below for the definition of SA; this result is due to Maddux [14] [15]) Finally, either (R6) # E or (R7) # E because of theorem 2.2 iv. Let us turn to polyadic equality algebras (PEA s) in place of cylindric algebras. We need the equality part because in polyadic algebras we cannot define the relation algebraic identity 1 # . 2. RELATION ALGEBRAS FROM CYL. ....

....AND POLYADIC ALGEBRAS 579 The following theorem is implicit in Monk [17] Ra # PEA 3 = R1 0 , R2 R7) 2.3) i.e. with the exception of the associativity axiom (R1) for ; all the axioms of RA are valid in Ra # PEA 3 . Motivated by the failure of associativity in Ra # CA 3 , Maddux [14] [15] introduced the weaker axiom (R1 0 ) called the semiassociativity law, see Definition 2.1 herein. He proved that (Ra # CA 3 ) # Mod (cycle law) R1 0 ) R2 R7) 2.4) He defined the variety SA of semi associative relation algebras as follows: SA def =Mod (R0) R1 0 ) R2 R7) ....

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R.D. Maddux. Some varieties containing relation algebras. Trans. Amer. Math. Soc., 272:501-- 526, 1982.

....a sentence of the loosely guarded fragment has a finite model, and thus the loosely guarded monodic fragment of predicate temporal logic and finite domains is decidable. 4. 2 Finite base property in algebraic logic The finite algebra on finite base property for weakly associative algebras [17] follows easily from the finite model property for the loosely guarded fragment. Let A = A; 0;1;1 , be a finite weakly associative algebra. Regard each a 2 A as a binary relation symbol. Then a relativised representation of A is a model of the following theory: 8xy[1 , x;y) x = ....

.... (r(x;y) s(x;y) for each r; s 2 A with r = s 8xy[r(x;y) s(y; x) for each r; s 2 A with r = s 8xy[1(x;y) r(x;y) 9z(s(x;z) t(z; y) for each r; s; t 2 A with r = s ; t 9xy r(x;y) for each r 2 A with r 6= 0: Every weakly associative algebra has a relativised representation [17]. It is easily seen that the conjunction of the above theory can be written as a loosely guarded sentence. Thus, by corollary 3.4, any finite weakly associative algebra has a finite relativised representation. One may also show in much the same way that WA has the finite base property: any ....

R Maddux, Some varieties containing relation algebras, Trans. Amer. Math. Soc. 272 (1982), 501--526.

....are studied in this section. Given a class of algebras, two kinds of operations can be considered in the study of its properties: basic operations (for the interpretation of the language) and non basic operations (defined in terms of basic operations, for instance in RA: domain and codomain [Mad82, Mad91a, DB93, FBHV93], symmetric quotients [BSZ86, BSZ89] residuals [JT52, JT93] Here, the non basic operations are: ffl domain and codomain (definition 2.12) ffl fork, cross and make a copy (definition 8.1) Depending on how they are used, properties can be classificated as basic (axioms and rules relating the ....

R. Maddux. Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272:501--526, 1982.

....n ary relations . We will prove the finite base property for several classes of algebras, including the classes Crs, polyadic Crs, WA, and locally cubic n dimensional relativized cylindric set al..gebras augmented with substitution operators. For definitions of these see section 3 and example 23, or [4,13]; but they all fit the framework described above, varying only in what functions are present and what properties of W are assumed. We actually prove a single result (theorem 4) that gives most of these as corollaries, and then extend it (theorem 24) to give the rest. The proof of theorem 4 starts ....

.... to each f 2 Sigma, and an existential sentence of the signature Sigma, such that K is the class of all n dimensional algebras A of signature Sigma such that (i) each f 2 Sigma is defined in A by ffi f , and (ii) A j= Example 23 The class WA of weakly associative algebras, defined in [13], is the closure under isomorphism of an EP class. For, it may be defined as the isomorphism closure of the class of all two dimensional algebras of signature Sigma = f Delta; Gamma; 0; 1; 1 0 ; g (1 0 is nullary, unary, and ; binary) where the defining sentence is (1 = 1) 1 = 1 0 ....

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R. Maddux. Some varieties containing relation algebras. Trans. Amer. Math. Soc. 272 (2), 1982,

....of these relations is given in Table 1; it is worth to point out that the table does not have an extensional interpretation, i.e. there is no RA whose composition is given by Table 1. Nevertheless, the base relations are the atoms of a semi associative relation algebra in the sense of Maddux [29]. Using the relation ECD, another RCC axiom can be written in algebraic form as follows: RCC 4 . a) C ECD# #NTPP (b) O ECD# #P Let V be the greatest relation over R. Notice, that the property ### #C#C# #R# V # R V (4.7) forces the algebraic part of a RCC model over R to be a complete BA ....

....the 9 intersection model of Egenhofer Herring [17] which is based only on topological properties. Another promising area of research is to consider the expressive power of relational structures more general than BRAs, for example, those, in which the associativity of the composition is relaxed [29]. Egenhofer Rodrguez [18] have given a spatial interpretation of such a structure. ....

Maddux, R. (1982). Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272, 501--526.

....construction for proving the completeness of first order logic is an important early example. The method is used by Roger Maddux [Maddux, 1978] to show that every semiassociative relation algebra has a representation as a subalgebra of a relativized proper relation algebra. Maddux went on in [Maddux, 1982] to show that every weakly associative relation algebra can be represented as a subalgebra of a proper relation algebra relativized to a symmetric, reflexive relation. See section 10 below. A considerable generalisation of this work on relativized relation algebra, also based on the step by step ....

....an atomic A network and all the other projections map N to a graph labelled everywhere by 0. Thus every atomic A network is essentially equivalent to an atomic A network for some . Notation For an atomic relation algebra A, let Net(A) be the class of all finite, atomic, A networks. 6 In [ Maddux, 1982 ] an atomic network is called an A labelling of ff where ff is the size of the matrix or equivalently the number of nodes in the graph. August 30, 1996 14 Comments 1. In the temporal reasoning literature [Allen, 1983, Villain and Kautz, 1986, Dechter et al. 1991] a network is considered as an ....

[Article contains additional citation context not shown here]

R Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501--526, 1982.

....(see [Lyn56] it is valid for nite relation algebras. These so called Lyndon conditions can be seen as approximations to RRA. The Lyndon conditions can be expressed naturally in terms of a two player game. This was done in [HH97d, HH97b] but much the same ideas were used by Maddux, in [Mad82] for example, and indeed in [Lyn50] We believe that the explicit use of games concentrates attention on the essential concepts rather than notational details, and so permits more lucid proofs. The game is played on networks . An (atomic) network is a nite complete directed graph, with each ....

....any atomic non associative algebra [Mad89] but we leave generalisations to the main part of the paper. Also, this is not Maddux s de nition, but it is equivalent: see proposition 31 below. 4 consider are like those given by Maddux as semantics for the weakly associative relation algebras [Mad82] The unit (the interpretation of the top element 1 in the representation) is a re exive, symmetric relation on the base set M of the representation, and all operations, in particular the composition ; are relativised to the unit. So a pair (x; y) of elements of M will stand in the composite ....

[Article contains additional citation context not shown here]

R D Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501{ 526, 1982.

....Henkin construction for proving the completeness of first order logic is an important early example. The method is used by Roger Maddux [Mad69] to show that every semi associative relation algebra has a representation as a subalgebra of a relativized proper relation algebra. Maddux went on in [Mad82] to show that every weakly associative relation algebra can be represented as a subalgebra of a proper relation algebra relativized to a symmetric, reflexive relation. See section 11 below. A considerable generalisation of this work on relativized relation algebra, also based on the step by ....

....and forms an atomic network. We see that an important assumption has been made, namely that an atomic network is consistent. This assumption turns out to be valid only on a certain subclass of the class of all relation algebras those where Net(A) forms a TAP class (defined later) 6 In [Mad82] an atomic network is called an A labelling of ff where ff is the size of the matrix or equivalently the number of nodes in the graph. 7 Network N is a tightening of network M if N and M have exactly the same nodes, and for all nodes i and j, N(i; j) M(i; j) November 30, 1995 11 2. Another ....

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R Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501--526, 1982.

....gener9 alizations of associative RAs. In these generalizations, the axiom of associativity for relational composition is dropped. This leads from (the so called) representable (RRA) to semiassociative (SA) weakly associative (WA) and non associative (NA) relational algebras. In 1982 Maddux [55] gave the following result: RRA ae RA ae SA ae WA ae NA. All these generalizations unfortunately deal only with one relational composition. Our equations for pseudo associativities given above show that the interplay of several relational compositions is essentially involved in relational ....

R.D. Maddux. Some varieties containing relation algebras. Transact. of the American Mathemtical Society, 272(2):501--526, 1982.

....j is always equivalent to 9x j 9x i ) It follows that the definable sets form an n dimensional cylindric algebra, and so we prove in theorem 11 the implication (3) 1) of theorem 1. In section 3, we introduce hyper networks and hyper bases. Hyper networks are very like the basic matrices of [Mad82, section 4] or the atomic networks of [HH97b] but as well as using atoms to label edges of these hyper networks, we also have labels for sequences of length greater than two. Hyper bases correspond approximately to Maddux s cylindric bases, the only difference being that the elements of a hyper basis are ....

....: a (l Gamma1) 4 Structures. If L is a signature and M an L structure, we write S M for the interpretation in M of a symbol S 2 L. For example, 1 0A is the identity of the relation algebra A. We usually identify (notationally) a structure with its domain. 2 Representation theory In [Mad82], it was shown that the weakly associative algebras are precisely those that have relativised representations in which the unit is a reflexive and symmetric relation. We will extend this to provide a representation theory for algebras in SRaCA n : the unit remains reflexive and symmetric, but the ....

R Maddux. Some varieties containing relation algebras. Trans. AMS, 272(2):501-- 526, 1982.

....relation algebras nor the class of completely representable cylindric algebras of any fixed dimension are elementary. 1 Introduction There are several types of representation for boolean algebras with operators, and of interest in this paper are the ordinary and complete representations [8, 10, 11, 17, 12, 13, 5, 1, 21, 19, 6]. Ordinary representations, or just representations , are isomorphisms from a boolean algebra with operators to a more concrete structure whose elements are sets, in which the boolean operators and Gamma are interpreted as [ and n and the other operators have certain set theoretically ....

.... A network (N) to be a function N : ff Theta ff At(A) for some set of nodes , ff, such that for all nodes l; m;n 2 ff, we have ffl N (l; m) Id if and only if l = m, ffl N (m; n) N (n; m) and ffl N (l; n) N (l; m) N (m; n) The network is said to be non empty if ff 6= Maddux [13] uses the term labelling instead of network , in the case where ff is an ordinal. Networks will be used as forcing conditions , and an atomic network can be thought of as an approximation to a complete representation of a relation algebra. Frequently, we do not have a special name for the set ....

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R Maddux. Some varieties containing relation algebras. Transactions of the AMS, 272(2):501-- 526, 1982.

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Maddux, R.D., Some Varieties Containing Relation Algebras, Transactions of the American Mathematical Society 272, 1982, pp501--526. 36

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Roger D. Maddux, Some varieties containing relation algebras, Transactions of the American Mathematical Society 272 (1982), 501-526.

....is a relation algebra if . The reduct #A, 0, 1# is a Boolean algebra, The reduct #A, 1 , # is a monoid, and, for all p, q, r # A, p;q) r =0i# (p;r) q =0i# (r;q)p=0. For more background on relation algebras and sequential algebras, we refer the reader to [5] [10], 15] 18] In, particular the definitions above can be restated in terms of equations, so both the class of all sequential algebras and the class of all relation algebras are varieties, denoted by RA and SeA respectively. Every relation algebra is (term equivalent to) a sequential algebra if ....

R. D. Maddux, Some varieties containing relation algebras, Trans. Amer. Math. Soc. 272:501-- 526, 1982.

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R D Maddux, Some varieties containing relation algebras, Transactions of the American Mathematical Society 272, 1982, 501--526.

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Maddux, R. (1982). Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272:501--526.

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R. Maddux, Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 501--526.

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R. Maddux, Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 501--526.

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Madd R. Maddux. Some varieties containing relation algebras. Trans. Amer. Math. Soc. 272 (2), 1982, 501--526.

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Ma2 R. Maddux, Some Varieties Containing Relation Algebras, Trans. AMS 272 (1982) 501--526.

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