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A Course in Universal Algebra

bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

Written for computer scientists, this volume offers a new model-theoretic approach to universal algebra and presents a systematic development of the methods of results of universal algebra that are useful in a variety of applications in computer science. The book is concerned with the algebraic characterization of axiomatic classes of algebras by closure operators generalizing the famous Birkhoff Variety Theorem and the algebraic characterization of related theories. The book also presents a study of term rewriting systems. Beside basic notions, the Knuth-Bendix completion procedure and terminal proof methods are considered. A third main topic is that of fixpoint techniques and complete ordered algebras. Algebraic specifications of abstract data types and algebraic semantics of recursive program schemes are treated as applications.

Universal algebra for computer scientists

A formalism for constructing and using axiomatic specifications in an arbitrary logical system is presented. This builds on the framework provided by Goguen and Burstall’s work on the notion of an institution as a formalisation of the concept of a logical system for writing specifications. We show how to introduce free variables into the sentences of an arbitrary institution and how to add quantitiers which bind them. We use this foundation to define a set of primitive operations for building specifications in an arbitrary institution based loosely on those in the ASL kernel specification language. We examine the set of operations which results when the definitions are instantiated in institutions of total and partial tirst-order logic and compare these with the operations found in existing specification languages. We present proof rules which allow proofs to be conducted in specifications built using the operations we define. Finally, we introduce a simple mechanism for defining and applying parameterised specifications and briefly discuss the program development

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Specifications in an arbitrary institution

This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.

Idempotent functional analysis: An algebraic approach

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

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Tensor products and bimorphisms

Equational compactness in equational classes of algebras

We show, for any subset system Z (as defined in Wright, Wagner, and Thatcher, T.C.S. 7 (1978), pp. 57–77) and any order preserving map $f:Q \to P$ of posets, the existence of a universal map $u_f :P \to P_f $ where $P_f $ is Z-complete and $u_f f$ is Z-continuous. This generalizes to arbitrary subset systems the result of Markowsky (T.C.S. 4 (1977), pp. 125–135) for chains, and the completions of Wright, Wagner, and Thatcher for union complete Z; our method, different from theirs, uses the time-honored direct construction of universal maps. Further, we obtain some results on the internal structure of $P_f $ with regard to Z-joins. Finally, we show that each element of the Z-completion of P is a Z-join of elements of P iff Z is union complete.

Completions of Partially Ordered Sets

Four formalizations of the instruction if–then–else are considered. Let K be an equational class of algebras. Assume the language has constants $ \bot $ in case II, T, F, $ \bot $ in case III and T in case IV. Any equational class with $ \bot $ is assumed to be strict, i.e. satisfies $F(\_, \bot ,\_) = \bot $ for all functions f Form the classes I $K^d $, II $K^{d \bot } $, III $K^t $ and IV $K^{t'} $ by adding an operation $[,,,]$ in cases I and II defined by \[ {\text{I}}\quad [x,y,z,w] = \left\{ \begin{gathered} z\quad {\text{if }}x = y, \hfill \\ w\quad {\text{else}} \hfill \\ \end{gathered} \right.\qquad {\text{II}}\quad [x,y,z,w] = \left\{ \begin{gathered} z\quad {\text{if }}x = y \ne \bot \hfill \\ w\quad {\text{if }} \bot \ne x \ne y \ne \bot , \hfill \\ \bot \quad {\text{else,}} \hfill \\ \end{gathered} \right. \]and by adding an operation $[,,]$ in cases III and IV defined by \[ {\text{III}}\quad [x,y,z] = \left\{ \begin{gathered} y\quad {\text{if }}x = T, \hfill \\ z\quad {\text{if }}x = F \hfi...

Evelyn Nelson

Papers

Tensor products and bimorphisms

Equational compactness in equational classes of algebras

Completions of Partially Ordered Sets

Equational bases for if-then-else

Boolean powers as algebras of continuous functions