Gunter Schmidt and Thomas Strohlein, Relation algebras: concept of points and representability, Discrete Mathematics 54 (1985), 83--92.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....To get the second theorem we use Monk s results in [Mo70] on the existence of completions of relation algebras. In x7 we define points, pairs, twins, functional elements, and identity atoms, and prove many things about them. That section contains generalizations of some theorems in [J82] and [SS85]. There is only one theorem in that section which holds for relation algebras but not for semiassociative relation algebras. This is exactly where to look to see why the algebra in (A) must be a relation algebra. In x8 we study point density and pair density. Highlights from this section: every ....

....algebra has an dimensional relational basis. The concluding x9 contains all the representation results discussed above. One of the corollaries obtained there is (H) A = ReP tA iff A is a simple complete point dense semiassociative relation algebra. This result generalizes Theorem 11 of [SS85] from relation algebras to semiassociative relation algebras. It also shows that one of the hypotheses of Theorem 11 is redundant. It is not necessary to assume that A is atomic. The authors of [SS85] state that their Theorem 11 simplifies the proof of a similar result given in a somewhat ....

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Gunter Schmidt and Thomas Strohlein, Relation algebras: concept of points and representability, Discrete Mathematics 54 (1985), 83--92.

.... on A (of type A A) For every two sets A; B, the relation A B = A B is called the universal relation (of type A B) For every two sets A; B, the relation A B = f(x; y) j false g is called the empty relation (of type A B) For the usual rules of the calculus of relations see [5, 8, 40, 41]. There are many di erent types of relations, however in this paper we shall use only two: total relations, and functions (univalent relations) The relation R A B is total if dom(R A B ) A. The relation R A B is a function (univalent relation) if 8x 2 A:8y; z 2 B: x; y) 2 R (x; z) 2 ....

G. Schmidt, T. Strolein. Relations algebras: Concept of points and representability. DISCR, 54:83-92, 1985.

.... Re ned designs of map algebra the arithmetic of dyadic relations , and related research, constitute the most traditional and lasting e ort to bridge rst order predicate reasoning with purely equational reasoning [17, 18] Several axiomatizations of map algebra are available (cf. e.g. 3] and [16]) our own, shown in Fig.1, is conceived with the aim of providing support to theorem proving activities based on a state of the art proof assistant, namely on Otter [12] Whatever version one chooses, the equalities in this xed initial endowment, describing the full variety of dyadic relations ....

....some of their consequences. The above lemma mainly relies on various elementary Boolean identities, and on some obvious consequences of the Peircean axioms (i.e. the logical axioms regarding ; 1 , and ) The only non obvious laws on maps needed are the socalled cycle law and Dedekind law (cf. [16]) shown in Fig.4 together with some consequences that are also needed (cf. Sec.4) the outline of the proof of the functionality lemma itself is shown in Fig.6, and will be commented on the end of this section. Unfortunately, the fact that a human being regards certain consequences of the axioms ....

G. Schmidt and T. Strohlein. Relation algebras: concepts of points and representability. Discrete Mathematics, 54:83-92, 1985.

.... C . For the first direction, it is sufficient to show that with the Dedekind rule, Q ; R v S implies Q ; S v R: assume Q ; R v S , then that is equivalent to Q ; R u S = and we have Q ; S u R v Q ; S u Q ; R) Conversely, assume that the Schroder equivalences hold. Then [ SS85 ] shows: Q ; R = Q u S ; R ) t (Q u S ; R ) R u Q ; S ) t (R u Q ; S ) Boolean lattice = Q u S ; R ) R u Q ; S ) t (Q u S ; R ) R u Q ; S ) t (Q u S ; R ) R u Q ; S ) t (Q u S ; R ) R u Q ; S ) join distributivity v (Q u ....

Gunther Schmidt and Thomas Strohlein. Relation algebras --- concept of points and representability. Discrete Math., 54:83--92, 1985. 27

....Introduction In 1941 Tarski [8] proposed a problem, that is, Is every relation algebra isomorphic to an algebra of all Boolean (ordinary) relations on a set . The positive answers of the question, called representation theorem for relation algebras, have been investigated. Schmidt and Strohlein [6, 7] gave a simple proof of the representation theorem for Boolean relation algebras satisfying Tarski rule and a point axiom. A representation theorem for fuzzy relation algebras satisfying a point axiom was proved by Kawahara and Furusawa [1] and categorical representation theorems of fuzzy ....

....u r ] ff) v r(fi u rff) r(fi u ff) rO = O by R4, A1(b) and R2(c) Hence rfi = rfi u r = rfi u (ff t fi) rfi u ff) t (rfi u fi) fi. 2 From proposition 2.4(a) 2.4(b) and 3. 4(c) the set of all crisp relations in fuzzy relation algebra R forms a (Boolean) relation algebra in the sense of [6]. 4 Ideal Relations J onsson and Tarski investigated ideal relations in Boolean relation algebras [9] In this section we define ideal relations [9] in fuzzy relation algebras, and consider some properties of ideal relations. Though in Boolean relation algebras ideal relations are just universal ....

G. Schmidt and T. Strohlein, Relation algebras : Concept of points and representability, Discrete Mathematics 54 (1985) 83--92.

....modern algebraic study of (binary) relations, namely relational calculus, was begun by Tarski; see [12] for details of the history of the study of Boolean relation algebras. In [10] Tarski proposed a formalisation of Boolean relation algebras and their representation problem. Schmidt and Strohlein [8, 9] gave a simple proof of a representation theorem for Boolean relation algebras satisfying the (so called) Tarski rule and a point axiom. Dedekind categories [2] or allegories [3] provide a categorical framework for relational calculus. In relational calculus one calculates with relations in an ....

....categories [2] or allegories [3] provide a categorical framework for relational calculus. In relational calculus one calculates with relations in an element free style, which makes relational calculus a very useful framework for the study of mathematics [7, 11] and theoretical computer science [8, 13] and also a useful tool for applications. Some element free formalisations of fuzzy relations and proofs of representation theorems were provided in [4, 5, 6] 1 2 Hitoshi Furusawa In this paper we consider relation algebras, which may not be Boolean, and provide their representation theorem. ....

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G. Schmidt and T. Strohlein, Relation Algebras: Concept of Points and Representability, Discrete Mathematics 54 (1985) 83--92.

....generated by symmetric and transitive relations, J onsson [Jon88] and Givant [Giv94] each provided such conditions. The readers refer the list of historical results of such a problem for Boolean relation algebras, which we introduced above, in chapter 2 of [BKS97] Schmidt and Strohlein [SS85] simplified the proof of the representation theorem by using a notion of point relations. They also showed the important role of point relations in computer science [SS93] Since it is comfortable for the author to use the notion of point relations concerning application of our relational ....

....are homogeneous ones on a fixed set X with values in the unit interval [0; 1] that is, functions R : X 2X [0; 1] The set of all such fuzzy relations on X constitutes a fuzzy relation algebra. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean and so the Schroder rule [SS85] an important axiom of Boolean relation algebras, does not work in our case. Instead of the Schroder rule we prefer to adopt the Dedekind formula. As fuzzy relations are naturally equipped with semi scalar multiplication by scalars in the unit interval, it 2 is necessary to add axioms F1 and F3 ....

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Schmidt, G. and Strohlein, T.: Relation Algebras: Concept of Points and Representability, Discrete Math. 54 (1985) 83--92.

....Zadeh s invention of the concept of fuzzy sets [19] Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relations, namely relational calculus, has been investigated since the middle of the nineteen century, see [13, 16, 17] for more details. Almost all modern formalisations of relation algebras are affected by the work of Tarski [18] Mac Lane [12] and Puppe [15] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which ....

G. Schmidt and T. Strohlein, Relation algebras : Concept of points and representability, Discrete Mathematics 54 (1985) 83--92.

....as heterogeneous or rectangular . A convenient framework to do so is given by category theory [1, 12, 13] Under certain circumstances, i.e. relational products exist or the point axiom is given, a relation algebra may be represented in the algebra Rel of concrete binary relations between sets [5, 6, 11, 13]. In other words, the algebra may be seen as an algebra of Boolean matrices. As known, not every (homogeneous or heterogeneous) relation algebra or Dedekind category need be representable and therefore need not be an algebra of Boolean matrices [1, 3, 4] In this paper, we will show that it is ....

G. Schmidt and T. Strohlein, Relation Algebras: Concept of points and representability, Discrete Mathematics 54 (1985) 83-92.

.... A (of type A A) For every two sets A; B, the relation A B = A B is called the universal relation (of type A B) For every two sets A; B, the relation A B = f(x; y) j false g is called the empty relation (of type A B) For the usual rules of the calculus of relations see [5, 8, 40, 41]. There are many di erent types of relations, however in this paper we shall use only two: total relations, and functions (univalent relations) The relation R A B is total if dom(R A B ) A. The relation R A B is a function (univalent relation) if 8x 2 A:8y; z 2 B: x; y) 2 R (x; z) 2 R ....

G. Schmidt, T. Strolein. Relations algebras: Concept of points and representability. DISCR, 54:83-92, 1985.

....Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated by Sanchez [11] and applied to medical models of diagnosis. On the other hand theory of relations, namely relational calculus, has a long history, see [8, 12, 13] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [14] Mac Lane [7] and Puppe [10] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, ....

.... (or graph grammars) A relational approach to set theory is studied by Kawahara [4] However the authors should mention that some ideas and results in [4] have been already given by [1] Concerning with applications to the relational theory of graphs and programs, Schmidt and Strohlein [12] gave a simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [13] on relations and graphs with many useful examples in computer science. The aim of the paper is to provide an algebraic formalization ....

[Article contains additional citation context not shown here]

G. Schmidt and T. Strohlein, Relation algebras : Concept of points and representability, Discrete Mathematics 54 (1985) 83--92.

....Goguen [2] generalized the concepts of fuzzy sets and relations taking values on partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [12] On the other hand theory of relations, namely relational calculus, has a long history, see [8, 13, 14] for more details on the history. Almost modern formalizations of relation algebras are affected by the work of Tarski [15] Mac Lane [7] and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, ....

....Lane [7] and Puppe [11] exposed a categorical basis for calculus of additive relations. Freyd and Scedrov [1] developed and summarized categorical relational calculus, which they called allegories. Concerning with applications to the relational theory of graphs and programs, Schmidt and Strohlein [13] gave a simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom. Also they wrote an excellent text book [14] on relations and graphs with many useful examples in computer science. Also Kawahara and Mizoguchi [3, 6, 4] developed relational ....

[Article contains additional citation context not shown here]

G. Schmidt and T. Strohlein, Relation algebras : Concept of points and representability, Discrete Mathematics 54 (1985) 83--92.

....[5] generalized the concepts of fuzzy sets and relations taking values from partially ordered sets. Fuzzy relational equations were initiated and applied to medical models of diagnosis by Sanchez [17] On the other hand, the theory of relations, namely relational calculus, has a long history, see [13, 18, 19] for more details. Almost all modern formalizations of relation algebras are affected by the work of Tarski [20] Mac Lane [12] and Puppe [16] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which ....

....Lane [12] and Puppe [16] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which they called allegories. Concerning applications to the relational theory of graphs and programs, Schmidt and Strohlein [18] gave a simple proof of a representation theorem for Boolean relation algebras satisfying the Tarski rule and the point axiom. They also wrote an excellent text book [19] on relations and graphs with many useful examples from computer science. In relational calculus one calculates with relations ....

[Article contains additional citation context not shown here]

G. Schmidt and T. Strohlein, Relation algebras : Concept of points and representability, Discrete Mathematics 54 (1985) 83--92.

....[1990b ] In a very preliminary draft J onsson [Draft] tentatively outlines another algebraic treatment of programs and program specification which is more extensively based on universal algebraic notions. In conclusion I list some references to other applications of the algebra of relations. Schmidt and Strohlein [1985] consider some requirements for relation algebra applied in the (relational) theory of graphs and programs. Maddux [1983] presents a sequent calculus for the calculus of relations, and Wadge [1975] and Hennessy [1980] develop natural deduction systems for the calculus of relations. Further ....

Schmidt, G., and Strohlein, T. [1985]. Relation Algebras: Concept of Points and Representability.

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