Engeler, E.(1988). A Combinatory Representation of Varieties and Universal Classes. Algebra Universalis, 24.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....e.g. combinatory algebras. A popular model of combinatory algebras is calculus [Barendregt, 1977] In this work we consider other models of combinatory algebras, namely graph models [Engeler, 1981A, Engeler, 1981B] It was shown that any algebraic structure can be embedded in a graph model [Engeler, 1988]. Hence graph models give rise to an algebraic model of computation in algebraic structures. So it appears appropriate to use them as models for symbolic computation. On the other hand, graph models have, similarly as analytic structures, a second facet. They are also endowed with the structure of ....

Engeler, E.(1988). A Combinatory Representation of Varieties and Universal Classes. Algebra Universalis, 24.

.... clear structure with respect to approximation as well to algebraization are Engeler graph models [Engeler, 1981A, Maeder, 1986] Graph models were already used as programming semantics for several mathematical structures, like varieties, geometries and analysis [Engeler, 1981B, Engeler, 1984, Engeler, 1988, Fehlmann, 1981, Seeland, 1978] We will show how to bring these graph models in a form where they allow the incorporation of the algebraic aspects of analysis as delivered by the differential fields theory. This goes back to an approach that was first outlined in [Engeler, 1990] Then ....

Engeler, E. (1988). A Combinatory Representation of Varieties and Universal Classes. Algebra Universalis, 24.

....and Dana S. Scott 1. E : ffl j Ea defines the strings of a s (including the empty string ffl) 2. E : a j bEb defines strings consisting either of the letter a alone or a string of n b s followed by an a followed by n more b s. 3. E : ffl j aa j EE defines strings of a s of even length. We may use the Fixed Point Theorem to provide a precise explanation of the semantics of these grammars. Since the operations X 7 ffflg[Xfag, X 7 fag[fbgXfbg, and X 7 ffflg[fagfag[XX are all continuous in the variable X , it follows from the Fixed Point Theorem that equations such as 1. X = ....

....in the notation of applicative algebra which has no variables a combination. Any combination F defines an n ary operation: F (x 1 ) x 2 ) Delta Delta Delta (x n ) What we have been remarking is that the algebras so obtained from combinations can be very rich. In a series of papers [Eng81, Eng] Engeler discussed just how rich these algebras can be. A representative result, following Engeler, will be exhibited here. Theorem 26 Given a signature (s 1 ; s 2 ; s n ) there are combinations F 1 ; F 2 ; F n defining operations on D of these arities such that whenever ....

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E. Engeler. A combinatory representation of varieties and universal classes. Algebra Universalis, ??:??--??, ?? To appear.

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