Peter Burmeister

Partial algebras—survey of a unifying approach towards a two-valued model theory for partial algebras

In this survey, with no proofs included, we collect some material scattered through recent papers and a planned monograph, which shows that partial algebras do have a two-valued first order model theory which is simpler and nicer than one might have expected it to be. In section 1 we comment and present some basic definitions. In section 2 a correct and complete two-valued first order logic is developed. In section 3 the three main concepts of “varieties” are presented, while sections 4 and 5 contain some additional axiomatizability results and some applications, respectively. Section 6 contains some additional remarks.

Partial Algebras — An Introductory Survey

Partial algebras are among the basic mathematical structures implemented on computers. Many-sorted algebras are basically partial algebras, too. These notes are meant to introduce into a theory of and a language for partial algebras in such a way that also a specification of (many-sorted) partial algebras as abstract data types can easily be performed. Besides the terminology and constructions from universal algebra (homomorphisms, generalized recursion theorem, epimorphism theorem, free partial algebras) also such from logic (existence equations and elementary implications), model theory (preservation and reflection of formulas by mappings) and from (elementary) category theory (factorization systems) prove to be quite useful for a good description of the arising concepts, as is shown at the end by the formulation of a “Meta Birkhoff Theorem”.

Algebraic transformation of unary partial algebras II: single-pushout approach

The transformation of total graph structures has been studied from the algebraic point of view over more than two decades now, and it has motivated the development of the so-called double-pushout and single-pushout approaches to graph transformation. In this article we extend the double-pushout approach to the algebraic transformation of partial many-sorted unary algebras. Such a generalization has been motivated by the need to model the transformation of structures which are richer and more complex than acyclic graphs and hypergraphs. The main result presented in this article is an algebraic characterization of the double-pushout transformation in the categories of all homomorphisms and all closed homomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization which may serve as a basis for implementation. Moreover, both categories are shown to satisfy the strongest of the HLR (High Level Replacement) conditions with respect to closed monomorphisms. HLR conditions are fundamental to rewriting because they guarantee the satisfaction of many rewriting theorems concerning confluence, parallelism and concurrency.

Quasi-Equational Logic for Partial Algebras

In the line of introducing a manageable model theoretic approach to partial algebras, here such classes of partial algebras are to be considered in which free algebras still exist (in a categorical language: which are epireflective). This note is to be understood as one among others introducing this kind of model theory (in another one, see [4], varieties of partial algebras are considered). We want to make an end to the widely spread opinion that there are several equational theories and several notions of validity around for partial algebras. At the same time we want to provide for all those who might use partial algebras such tools that they hopefully can really work with. We just use the usual first order formulas of a model theoretic language with terms but substituting the notion of “equation” by that of “existenceequation”, and we intend to give a procedure how to interpret their satisfaction and their validity in partial algebras. In this note we shall restrict ourselves to existence-equations and quasi-existence-equations (the latter comparable to the notion of quasi-equation in [8] §11.1., i.e. essentially: universally quantified Horn-formulas).

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let Uf denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some Uf-algebra. For any U-algebraAsetUA=UiεI({i}×(AKi—domfiA)), where (Ki)iεI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, Uf) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ⊆N such that there exists a bijective mapping σ:UF(N, U) →N −M satisfying σ((i, α)) ∉ α(Ki) for all (i, α) ∈UF (N, U), and, for the structureg=(gi)iεI defined by ,gi: =fiF(N, U) ∪ {(α, σ((i, α))) | (i, α ∈UF(N, U)} idM induces an isomorphism betweenF(M, Uf), and (F(N, U)g).

THE MEANING OF BASIC CATEGORY THEORETICAL NOTIONS IN SOME CATEGORIES OF PARTIAL ALGEBRAS. II PRODUCTS AND COPRODUCTS

This paper is a direct continuation of the paper [4], to which we refer the reader for the definitions of basic concepts and some preliminary results. Observe that also the numeration of the statements and figures is continued. For concepts not defined here we refer the reader in particular to [4] and [2], however see also [1] or [3] as far as partial algebras are concerned, and [5] or [7] with respect to category theoretical concepts. A great part of the results and proofs presented here can already be found in the report [8], which has never been published so far.

On maximal objects in classes of (DI)graphs determined by prescribed factorobjects

In connection with the papers [1] and [2] of Nesetril, Pultr and Vinarek we characterize the maximal objects in classes of graphs and digraphs which are determined by prescribed factorobjects. For this purpose a special quasi-ordering on the set of the given factorobjects is introduced and investigated.