E. G. Manes, Algebraic Theories, Graduate Texts in Mathematics, Vol. 26, Springer-Verlag, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....parallel connection, which we might denote Y , as opposed to the true concurrent connection that we have denoted X k Y and for which we have given a universal characterision above. It is interesting to note that forming the so called Lawvere algebraic theory (in the sense of [28] and [29]) of a hidden sorted speci cation gives rise to a structure generalising some traditional structures of process algebra, in that there is an action of the monoid of all method expressions (in the sense of FOOPS [21] on the collection of all attribute expressions (again in the sense of FOOPS) ....

Ernest Manes. Algebraic Theories. Springer, 1976. Graduate Texts in Mathematics, Volume 26.

.... and final coalgebras from the programming perspective and to the categorical approach to functional programming in general, we refer to [Fok92, BdM97] The recursion and corecursion schemes used in the paper are described in [UV99, UVP01] The classic category theory texts treating (co)monads are [Man76, BW84] Throughout the paper, we work in one base category C about which we do not make any specific assumptions other than the existence of the particular coproducts, initial algebras etc. that we name. The category Set of sets and set theoretic functions is always a possible choice for C . The ....

.... monads and substitution As matter of fact, every monad M on Set has something to do with substitution in the usual sense of term substitution: MA may be understood as the free (i.e. term equivalence class) algebra over A for some (possibly infinitary) signature and collection of equations [Man76, Sec. 1.5] This and further related theory, however, remains outside of the scope of this paper. A practical conclusion is that a monad may always be thought of as a type constructor endowed with a substitution like operation. The monad laws state basic properties of substitution. One might say ....

E. G. Manes. Algebraic Theories, vol. 26 of Graduate Texts in Mathematics. Springer-Verlag, 1976.

.... theory L# L # , such that, if L # is the enriched Lawvere theory associated with side e#ects, the new enriched Lawvere theory corresponds to T (S ) S , where L corresponds to T The answer is yes, it is remarkably natural, and, in various guises, forms of it have existed since the 1960 s [4, 15], and is known as the tensor product of theories. It simply amounts to taking the operations of both theories and demanding that they commute with each other, while retaining the equations of both: for instance, in the case where S = V Loc , combining state with nondeterminism, if there were ....

E. G. Manes, Algebraic Theories, Graduate Texts in Mathematics, Vol. 26, New York: Springer-Verlag, 1976.

....to adjunctions, probably the most pervasive notion in CT. The connection between monads and adjunctions was established independently by Kleisli and Eilenberg Moore in the sixties. Monads, like adjunctions, arise in many contexts (e.g. in algebraic theories) Good CT references for monads are [Man76,BW85,Bor94] It is not surprising that monads arise also in applications of CT to Computer Science (CS) We will use monads for giving denotational semantics to programming languages, and more specifically as a way of modeling computational types [Mog91] to interpret a programming ....

....it seems natural that a collection should have only a finite number of elements, and there should be an empty collection ; TA and a way of merging two collections using a binary operation : TA TA TA. There are several equivalent definitions of monad (the same happens with adjunctions) Man76] gives three definitions of monad triple called: in monoid form (the one usually adopted in CT books) in extension form (the most intuitive one for us) and in clone form (which takes composition in the Kleisli category as basic) We consider only triples in monoid and extension form. Notation ....

E. Manes. Algebraic Theories. Graduate Texts in Mathematics. Springer, 1976.

.... For each function f 2 [D P[E ] we define the extension of f , denoted f , where f 2 [P[D] P[E] by: f (A) x2A f(x) For each powerdomain constructor P[ Gamma] the structure (P[D] j D (x) is a Kleisli triple or, alternatively, a monad, and thus satisfies the Kleisli triple laws [27]: j D = id P[D] f = f ffi j D g ffi f = g ffi f) This provides a canonical way of composing the semantics of programs. In the particular setting of the denotations of commands, it is worth noting that JC 1 ; C 2 K would be given by: JC 1 ; C 2 K = JC 2 K ffi JC 1 K 3.3. Pers on ....

Manes, E.: 1976, Graduate Texts in Mathematics, Vol. 26. Springer-Verlag.

....whereas the opens do not. A problem, that was solved by Johnstone, is whether frames in general can be presented by semilattices. For nitary algebraic theories presentations always present algebras. For an overview of the method of presenting algebras from a set of generators and relations see [Man76] or [Vic89] This is not always the case for innitary algebraic theories (see [Joh82] For frames, however, it is the case, as shown in [Joh82] In the icoverage theoremj Johnstone gives an explicit description of a frame being presented from a set of generators and relations. In this way, he ....

E.G. Manes. iAlgebraic Theoriesj, Graduate Texts in Mathematics 26, SpringerVerlag, 1976.

....to adjunctions, probably the most pervasive notion in CT. The connection between monads and adjunctions was established independently by Kleisli and Eilenberg Moore in the sixties. Monads, like adjunctions, arise in many contexts (e.g. in algebraic theories) Good CT references for monads are [Man76, BW85, Bor94] It is not surprising that monads arise also in applications of CT to Computer Science (CS) We will use monads for giving denotational semantics to programming languages, and more specifically as a way of modeling computational types [Mog91] to interpret a programming ....

....it seems natural that a collection should have only a finite number of elements, and there should be an empty collection ; TA and a way of merging two collections using a binary operation : TA TA TA. There are several equivalent definitions of monad (the same happens with adjunctions) Man76] gives three definitions of monad triple called: in monoid form (the one usually adopted in CT books) in extension form (the most intuitive one for us) and in clone form (which takes composition in the Kleisli category as basic) We consider only triples in monoid and extension form. Notation ....

E. Manes. Algebraic Theories. Graduate Texts in Mathematics. Springer, 1976.

.... 35] A uniform treatment of the model theory of classical equational logic is now possible due to the comprehensive development of categorical universal algebra; without any claim of completeness, I mention the so called Lawvere algebraic theories, either in classical form [69] or in monadic form [70] (although neither of these fits order sorted algebra 12 Devoted to modularisation issues. 13 The precise definition is given in Chapter 5. 14 Sometimes called language or vocabulary in classical logic textbooks. 15 As opposed to to the syntactic perspective that regards terms as tree like ....

Ernest Manes. Algebraic Theories. Springer, 1976. Graduate Texts in Mathematics, Volume 26.

....parallel connection, which we might denote X Omega Y , as opposed to the true concurrent connection that we have denoted X k Y and for which we have given a universal characterision above. It is interesting to note that forming the so called Lawvere algebraic theory (in the sense of [27] and [28]) of a hidden sorted specification gives rise to a structure generalising some traditional structures of process algebra, in that there is an action of the monoid of all method expressions (in the sense of FOOPS [20] on the collection of all attribute expressions (again in the sense of FOOPS) ....

Ernest Manes. Algebraic Theories. Springer, 1976. Graduate Texts in Mathematics, Volume 26.

.... A uniform treatment of the model theory of classical equational logic is now possible due to the comprehensive development of categorical universal algebra; without any claim of completeness, I mention the so called Lawvere algebraic theories, either in classical form [33] or in monadic form [34] (although neither of these fits order sorted algebra nicely) the theory of sketches [2] and the recently developed theory of abstract algebraic institutions [37, 38] However, no uniform proof theory has previously been developed for all these equational logics. It could be argued that, at ....

Ernest Manes. Algebraic Theories. Springer, 1976. Graduate Texts in Mathematics, Volume 26.

....the target category of the forgetful functor, from SET to the category SET S of all S indexed sets to get substitution systems with many sorted, or order sorted, infinite terms as morphisms. Constructions of this kind have been rather thoroughly explored in category theory; for example, see [33]. 7.2 Fixpoint Equations Fixpoint equations are used in computer science to define many different structures, and least fixpoints are particularly used, because their existence can often be guaranteed by the well known Tarski fixpoint theorem. A general context for these considerations is a ....

Ernest Manes. Algebraic Theories. Springer, 1976. Graduate Texts in Mathematics, Volume 26.

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E. G. Manes, Algebraic Theories, Graduate Texts in Mathematics, Vol. 26, Springer-Verlag, 1976.

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E. G. Manes, Algebraic Theories, Graduate Texts in Mathematics, Vol. 26, New York: Springer-Verlag, 1976.

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E. G. Manes, Algebraic theories, Graduate Texts in Mathematics 26. Springer-Verlag, New York, 1976.

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E. G. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, Berlin, 1976.

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Ernest G. Manes. Algebraic Theories, Graduate Texts in Mathematics 26. Springer Verlag, 1976.

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