R. McKenzie, The representation of relation algebras, Ph.D. thesis, University of Colorado at Boulder, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some special representations exist for these algebras. Therefore Lyndon knew that every small relation algebra is representable on a set of cardinality either or else not more than thirteen. Some explicit representations for simple small relation algebras were given in R. McKenzie s dissertation [McK66], pp. 38 40, and in F. Backer s seminar report [B70] pp. 11 20. A simple 8 element relation algebra was missed in the latter survey, and an 8 element direct product of two simple relation algebras was omitted from the former. B70] contains a representation on infinitely many elements for the ....

....h, defined by h(a i ) R i for all i n, has a unique extension to an embedding H of A into ReU . Conversely, if A is representable, then there is an isomorphism H from A into ReU for some set U , and hU; H(a 0 ) H(an Gamma1 )i is a model of Th(A) A similar construction is used in [McK66], pp. 108 109, and also in [J82] McKenzie and J onsson use a relation symbol for every element of the algebra. Their method therefore works even for infinite relation algebras, but for finite ones the resulting theory is exponentially larger than the one built here. This connection between ....

R. N. W. McKenzie, The representation of relation algebras, Doctoral dissertation, University of Colorado, Boulder, 1966.

....further investigations . Peirce was not so enthusiastic, as shown by his brief historical summary. Schroder developed Peirce s algebra from Note B, but Peirce thought that Professor Schroder attaches, it seems to me, too high a value on this alge [Lyndon1956] p. 307, footnote 13. See [McKenzie1966] pp. 37 39, or [McKenzie1970] p. 286. Tarski1941] p. 74. bra. Nevertheless, Schroder s work deserves much more attention than it has been given. 2) It was noted above that Lowenheim s Theorem, now conceived as the first theorem in model theory, was a theorem about the calculus of ....

Ralph N. W. McKenzie, The representation of relation algebras, Doctoral dissertation, University of Colorado, Boulder, 1966, pp. vi+128.

....This means that RRA is the closure of CRA under taking subalgebras. Hence, because RRA is a variety, it is the variety generated by CRA. We can also observe in passing that RRA is canonical (closed under the map A 7 A ) Monk s proof is unpublished (his result was reported in [McK66, p. 66] but several proofs are now known. In [HH97d] a proof by saturation was sketched. On the other hand, the notion of complete representation is de nitely stronger than the original one, even for atomic algebras. CRA itself is not elementary [HH97b] and since RRA is elementary, it ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....dimension of G. Using his construction, Lyndon obtained an integral nonrepresentable relation algebra with 8 atoms from a projective line with 6 points. The next reduction in size for the smallest known integral nonrepresentable relation algebra was accomplished by McKenzie. Theorem 20. McKenzie [17, 18]) There are at least 60 integral relation algebras with 4 atoms, and one of them is not representable. Equation (M) is valid in RRA but fails in McKenzie s example. This was not McKenzie s proof. x 01 (x 02 x 03 ; x 32 ) x 21 x 24 ; x 41 ) M) x 03 ; x 30 ; x 01 x 32 ; x 21 ) ....

Ralph N. W. McKenzie, The representation of relation algebras, Ph.D. thesis, University of Colorado, Boulder, 1966, Doctoral dissertation, pp. vi+128.

....the preservation of composition. 2. If A is simple then any non zero homomorphism can only be an isomorphism. Thus N is a homomorphism if and only if it is a representation. The following theorem is due to J. D. Monk. Here we give an alternative, simpler proof than the original one (reported in [McKenzie, 1966], theorem 2.12) THEOREM 22 (J. D. Monk) A relation algebra A is representable if and only if its ultrafilter extension A has a complete representation. 14 PROOF: Since A can be identified with a subalgebra of its ultrafilter extension, any representation of the extension automatically ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....2 This means that RRA is the closure of CRA under taking subalgebras. Hence, because RRA is a variety, it is the variety generated by CRA. We can also observe in passing that RRA is canonical (closed under the map A 7 A ) Monk s proof is unpublished (his result was reported in [McK66, p. 66] but several proofs are now known. In [HH97d] a proof by saturation was sketched. On the other hand, the notion of complete representation is de nitely stronger than the original one, even for atomic algebras. CRA itself is not elementary [HH97b] and since RRA is elementary, it ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....of composition. 2. If A is simple then any non zero homomorphism can only be an isomorphism. Thus N is a homomorphism if and only if it is a representation. 2 The following theorem is due to J. D. Monk. Here we give an alternative, simpler proof than the original one (reported in [McK66] theorem 2.12) November 30, 1995 31 THEOREM 22 (J. D. Monk) A relation algebra A is representable if and only if its ultrafilter extension A has a complete representation. 14 PROOF: Since A can be identified with a subalgebra of its ultrafilter extension, any representation of the ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....map. So (xx; yz) 2 E n Gamma1 , as required. 2 The following two theorems show (4) 2) of theorem 1, by applying theorem 46 to A . Theorem 45 If A has an n smooth relativised representation then A has a complete n smooth relativised representation. cf. Monk s theorem (reported in [McK66], theorem 2.12) that if a relation algebra A is representable then its canonical extension A has a complete representation. Proof [sketch] 25 To say that A has an n smooth relativised representation is to say that the firstorder theory Sm n , defined in definition 41 part 3, is ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....representation X, we have X(1) ff D where D is the domain. A simple, representable cylindric algebra always has a cubic representation. 2. 3 Languages For both relation and finite dimensional cylindric algebra there is a very natural, first order language corresponding to an algebra (see [McK66] Let A be any relation algebra, and let L = L(A) be the first order language with one binary predicate symbol for each element of A. We use the same symbol for an element of A as for the corresponding binary predicate in L(A) This will not lead to ambiguity: for a 2 A, if we write a(x; y) we ....

....of composition in L and the definition of a representation, that a network with nodes a; b; c; d with d c; d b; d]a; c]b (below) must embed in any representation of L. 3 The integral relation algebras possessing 1 transitive representations are called permutational relation algebras in [McK66] March 18, 1997 17 u u u u u Gamma Gamma Gamma Gamma Gamma a b c d Now the map : a 7 b; b 7 a; c 7 c is a local isomorphism as each distinct pair from a; b; c are related by #. However cannot extend to an automorphism of the representation as d would have to map to a ....

R McKenzie. The representation of relation algebras. PhD thesis, University of Colorado at Boulder, 1966.

....and showed that unitary tabular allegories were complete with respect to set theoretic relations. ffl Backhouse and his colleagues have investigated an extensive theory of datatypes, based on the calculus of relations [1] ffl Relation algebras have been further investigated by McKenzie in [66, 67] and Maddux in [62, 63] Maddux also wrote an interesting paper about the history of binary relations [64] which is recommended for further reading. ffl The relational language Ruby has been used for designing hardware, for example see the work of Jones, Sheeran and Hutton [90, 49, 91, 45, 50] ....

Ralph N. W. McKenzie. The representation of relation algebras. PhD thesis, University of Colorado, 1966.

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R. McKenzie, The representation of relation algebras, Ph.D. thesis, University of Colorado at Boulder, 1966.

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R. McKenzie, The representation of relation algebras, Ph.D. thesis, University of Colorado at Boulder, 1966.

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