The classes in Valiant's theory are classes of polynomials defined by arithmetic circuits. We characterize them by different notions of tensor calculus, in the vein of Damm, Holzer and McKenzie. This characterization underlines in particular the role played by properties of parallelization in these classes. We also give a first natural complete sequence for the class VPnb, the analogue of the class P in this context.

Free algebras with an arbitrary number of free generators in varieties of BL-algebras generated by one BL-chain that is an ordinal sum of a finite MV-chain Ln, and a generalized BL-chain B are described in terms of weak Boolean products of BL-algebras that are ordinal sums of subalgebras of Ln, and free algebras in the variety of basic hoops generated by B. The Boolean products are taken over the Stone spaces of the Boolean subalgebras of idempotents of free algebras in the variety of MV-algebras generated by Ln.

2000 Mathematics subject classification: primary 03G25, 03B50, 03B52, 03D35, 03G25, 08B20.

Recursively presented topological spaces are topological spaces with a recursive system of basic neighbourhoods. A recursively enumerable (r.e.) open set is a r.e. union of basic neighbourhoods. A set is everywhere r.e. open if its intersection with each basic neighbourhood is r.e. Similarly we define everywhere creative, everywhere simple, everywhere r.e. non-recursive sets and show that there exist sets both with and without these everywhere properties.

Notions of effective complementation in effective topological spaces are considered, and several types of non-complemented sets are constructed. While there are parallels with recursively enumerable sets, some unexpected differences appear. Finally, a pair of splitting theorems is proved.

This paper attempts to classify the least ordinal α0 for which E(α0) (the E closure of α0 ∪ {α0}) is inadmissible. Among the result proved are (i)Lα0 = ZFC-; (ii)α0 is very large in comparison with the least ordinal satifying (i); (iii) (α0, α] marks precisely an ω-Gap, where α¯ = E(α0) ∩ ON; (iv) the Kr-sequence of α0 has length ω.

An infinite collection of indecomposable isols such that no isol is comparable to certain infinitary combinations of the others is constructed, extending a result of Dekker and Myhill. This collection is then used to investigate differences between the arithmetic of classical RET's and that of RET's on recursive manifolds, a difference relevant to the manifold equivalent of the Schröder-Bernstein Theorem.

In 1981 two notions of effective presentation of countable connected graphs were formulated by J. C. E. Dekker—namely, edge recognition algorithm graphs and minimal path algorithm graphs. In this paper we show that every planar graph has a minimal path algorithm presentation but that some graphs have no minimal path algorithm presentations. We introduce the notion of a shortest distance algorithm graph, show that it lies strictly between the two notions of Dekker, and show that every graph has a shortest distance algorithm presentation. Finally, in contrast to Dekker's result about minimal path algorithm graphs, we produce a shortest distance algorithm graph which has no spanning tree which is an edge recognition algorithm graph.

We examine the concepts of nowhere simplicity in a wide class of abstract dependence systems. Initially we examine how many of the existing results valid for L(ω), the lattice of r.e. sets, have analogues valid for more general lattices. For example, we show that any r. e. subspace of V∞ can be decomposed into a pair of nowhere simple subspaces.

For Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).

For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.

1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.

2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.

3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.

Let the word “graph” be used in the sense of a countable, connected, simple graph with at least one vertex. We write Qn and Ocn for the graphs associated with the n-cube Qn and the n-octahedron Ocn respectively. In a previous paper (Dekker, 1981) we generalized Qn and Qn to a graph QN and a cube QN, for any nonzero recursive equivalence type N. In the present paper we do the same for Ocn and Ocn. We also examine the nature of the duality between QN and OcN, in case N is an infinite isol. There are c RETs, c denoting the cardinality of the continuum.

An infinite subset of ω is monotone (1–1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1–1 on it). A recursively enumerable set is co-monotone (co-1–1) just if its complement is monotone (1–1). It is shown that no implications hold among the properties of being cohesive, monotone, or 1–1, though each implies r-cohesiveness and dense immunity. However it is also shown that co-monotone and co-1–1 are equivalent, that they are properly stronger than the conjunction of r-maximality and dense simplicity, and that they do not imply maximality.

It is shown that no functor F exists from the category of sets with injections, to the category of algebraically closed fields of given characteristic, with monomorphisms, having the properties that for all sets A. F(A) is an algebraically closed field having transcendence base A and for all injections f. F(f) extends f. There does exist such a functor from the category of linearly-ordered sets with order monomorphisms.

An application to model-theory using the same methods is given showing that while the theory of algebraically closed fields is ω-stable, its Skolemization is not stable in any power.