Partial Isomorphisms Research Articles

Extending partial isomorphisms of graphs

Let X be a finite graph. Then there exists a finite graph Z containing X as an induced subgraphs, such that every isomorphism between induced subgraphs of X extends to an automorphism of Z.

A representation theorem for languages with generalized quantifiers through back-and-forth methods

We obtain in this paper a representation of the formulae of extensions ofL ωω by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraisses theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.

Higher Education as a Range of Educational Activities for Student Teachers

Summary Taking German experience as her point of departure, the author pleads for a practice‐based reform of secondary teacher education in universities by counter‐balancing the traditional overaccentuation of theory and abstract cognitive learning. A broader concept of practice and the suggestion of a ’partial isomorphism’ of universities and schools lead to the definition of higher education courses as practice and as fields of educational activities. In training seminars, student teachers have particular opportunities of profession‐oriented experiences and insights. If they learn, from the very beginning of their studies, to interpret the seminar sessions as learning situations and to organise them deliberately in cooperation with the seminar leaders, they can build the first elements of professional competence. The article reports how this conception of teacher education as field of, and preparation for, practice has been realised in an introductory seminar for freshmen at the Ruhruniversitat Bochum.

A hough transform technique for subgraph isomorphism

The Generalized Hough Transform is an established technique for geometric shape matching. An analogous technique is presented here for approximate solution of subgraph isomorphism problems. Its chief advantages are that it can also be used to find partial isomorphisms and that it can be readily implemented in parallel on a network of simple processors.

Different concepts of ‘‘ray’’ in optics: link between resolving power and radiometry

Of the many concepts of light rays the most important are the ancient one of Euclid and a more recent one as described by Newton. Euclid’s rays do not carry energy but only describe potentialities for vision of detail. They may be said to carry structural information. Newton’s rays transport energy and form the basis for radiometry. Geometry yields relations dealing with the multitude of rays. In Newton’s interpretation these are radiometric relations, whereas in Euclid’s interpretation these relations deal with resolving power. Thus the partial isomorphism between formulas of radiometry and the theory of resolving power of optical instruments results from simple geometrical relations between line congruences.

A group-theoretical invariant for elementary equivalence and its role in representations of elementary classes

There is a natural map which assigns to every modelU of typeτ, (U e Stτ) a groupG (U) in such a way that elementarily equivalent models are mapped into isomorphic groups.G(U) is a subset of a collection whose members are called Fraisse arrows (they are decreasing sequences of sets of partial isomorphisms) and which arise in connection with the Fraisse characterization of elementary equivalence. LetECλU be defined as {U eStrτ: ℬ ≡U and |ℬ|=λ; thenEGλU can be faithfully (i.e. 1-1) represented onto G(U) ×π*, whereπ*, is a collection of partitions over λ∪λ2∪....

Partial isomorphisms and intuitionistic logic

A game for testing the equivalence of Kripke models with respect to finitary and infinitary intuitionistic predicate logic is introduced and applied to discuss a concept of categoricity for intuitionistic theories.

A characterization of hypercubes

A combinatorial characterization of hypercubes is given. A hypercube is defined as the undirected Hasse diagram of a Boolean lattice. The method used is inductive extension of partial isomorphisms.

Categoricity regained

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by(i) enlarging our language so as to allow expressions of infinite length, and(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that(i) if f ∈ P then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and(ii) if f ∈ P, a ∈ A, and b ∈ B then there exist g,h ∈ P, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .

Congruence-free inverse semigroups with zero

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).

Partial Isomorphisms and Infinitary Languages

Partial Isomorphisms and Infinitary Languages

The Co-Simple Isols

Let E {0, 1, .. }, E* {O, ? 1, ...I}, P(E) = the class of all subsets of E, XkE the Cartesian product of k copies of E. A partial isomorphism is a 1-1 partial recursive function defined on a subset of E. If a, Se P(E), a and , are recursively equivalent (a 2? S) if there is a partial isomorphism p with a ' domain p and p(a) = 6; the equivalence class of a under this relation is its recursive equivalence type (r.e.t.) and the collection of all r.e.t.'s is denoted by t7. The r.e.t. of an isolated (i.e., finite or immune) set is an isol; A denotes the collection of isols, and A* the isolic integers (ring of differences of isols). These systems were introduced in [3], where various subsystems were also considered. Among these are i from this it follows that the results of ?? 2-3 of [7] remain valid when A* is replaced by A*. We are able to prove only a weak

Isomorphous replacement in a copolyamide system

AbstractIn a study of isomorphic behavior, it was found that the melting point—composition curve of the copolyamides prepared from caprolactam and p‐aminomethylcyclohexylcarboxylic acid was sigmoidal, indicating that the methylcyclohexyl group was interchangeable in the crystal lattice with aliphatic carbon atoms of ω‐aminocaproic acid. On the other hand, the benzyl group in the copolymers of caprolactam and p‐aminomethylbenzoic acid did not interchange as well, and only partial isomorphism occurred. The analogous meta derivatives, m‐aminomethylcyclohexylcarboxylic acid and m‐aminomethylbenzoic acid, did not fulfill the spatial requirements of caprolactam. Although V‐shaped melting point—composition curves resulted, the disturbance in the crystal network was more severe with the m‐phenyl derivative than with the m‐cyclohexyl derivative.

A partial isomorphism between the functions of Lucas and Weierstrass

A partial isomorphism between the functions of Lucas and Weierstrass