Partial Homomorphisms Research Articles

On the Endomorphism Monoid of Certain Ultrametric Speces

In this work we investigate the endomorphism monoid of certain ultrametric spaces. According to our main result, if X =<X,e> is an ultrametric space such that the range of e is finite, then the set of locally finite endomorphisms is dense in the endomorphism monoid of X and the endomorphism monoid of X has a dense, locally finite submonoid. This can be regarded as a homomorphism oriented counterpart of some ecently obtained results about the existence of dense, locally finite subgroups of the automorphism group of certain homogeneous structures. Further, as a byproduct, we obtain Hrushovski style extension theorems for the ages of certain ultrametric spaces, but here, instead of partial isomorphisms we extend partial homomorphisms.

Counterpart Semantics for a Second-Order μ-Calculus

Quantified μ-calculi combine the fix-point and modal operators of temporal logics with (existential and universal) quantifiers, and they allow for reasoning about the possible behaviour of individual components within a software system. In this paper we introduce a novel approach to the semantics of such calculi: we consider a sort of labeled transition systems called counterpart models as semantic domain, where states are algebras and transitions are defined by counterpart relations (a family of partial homomorphisms) between states. Then, formulae are interpreted over sets of state assignments (families of partial substitutions, associating formula variables to state components). Our proposal allows us to model and reason about the creation and deletion of components, as well as the merging of components. Moreover, it avoids the limitations of existing approaches, usually enforcing restrictions of the transition relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of. The paper is rounded up with some considerations about expressiveness and decidability aspects.

A BIRKHOFF TYPE THEOREM FOR STRONG VARIETIES

Algebraic systems with partial operations have different ways to interpret equality between two terms of the language. A strong identity is a formula which says that two terms are equal in the algebra if the existence of one of them implies the existence of the other one and in the case of existence their values are equal. A class of partial algebras defined by a set of strong identities is called a strong variety. In the characterization of strong varieties in the case of partial algebras by means of a Birkhoff-type theorem there appeared a new concept, regularity of partial homomorphisms and partial subalgebras. Here we define and study these operators from two different perspectives. Firstly, in their relation with other well known concecpts of partial homomorphisms and partial subalgebras, as well as with the po-monoid of Pigozzi for the H, S and P operators. Secondly, in regard to the preservation of the different types of formulae that represent equality in the case of partial algebras for these operators. Finally, we give a characterization of the strong varieties as classes closed under regular homomorphisms, regular subalgebras, direct products and that satisfy a closure condition.

SINGLE-PUSHOUT TRANSFORMATION OF TOTAL ALGEBRAS

We characterize the pairs of partial homomorphisms of total Σ-algebras that have a pushout in the corresponding category, for an arbitrary signature Σ. This characterization provides the application condition for the single-pushout approach to the transformation of total algebras.

Computable and Continuous Partial Homomorphisms on Metric Partial Algebras

AbstractWe analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations.

On Definability of Hypergraphs by Their Semigroups of Homomorphisms

In studying mathematical objects on the basis of their mapping systems considerable attention is given to the endomorphism semigroups and the partial endomorphisms semigroups of the investigated objects. It is natural to apply the same attention to systems of partial homomorphisms of these objects. However, such systems were rarely considered, since in these situations the usual composition of binary relations cannot be used and it is necessary to consider such mapping systems together with other operations. Experience shows that it is possible to consider the operations of right restrictive multiplication and left restrictive multiplication introduced by Wagner in [14] in the setting of operations for systems of partial homomorphisms. For example, Schein [11] and Molchanov [8], [9] showed that in individual cases systems of partial homomorphisms endowed with the restrictive multiplications have simple structure and contain important information about the investigated objects. This paper continues investigations in this direction. We describe here a wide class of hypergraphs which are determined up to isomorphism by their restrictive semigroups of partial homomorphisms. We also apply this result in analysing some problems of endomorphism semigroups of hypergraphs. The main result of the paper — theorem 3.1 — was announced in [7]. Note also that results of this paper generalize and correct a series of statements formulated without proofs in the end of the survey article [10]. The author wishes to thank Professor Boris M. Schein and the referee of this paper for their numerous valuable suggestions.

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

The single-pushout approach to graph transformation is extended 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 graphs and hypergraphs. The main result presented in this article is an algebraic characterization of the single-pushout transformation in the categories of all conformisms, all closed quomorphisms, and all closed-domain closed quomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization that may serve as a basis for implementation. Moreover, all three categories are shown to satisfy all of the HLR ( high-level replacement) conditions for parallelism, taking as occurrences the total morphisms in each category. Another important result presented in this article is the definition of HLR conditions for amalgamation, which are also satisfied by the categories of partial homomorphisms considered here, taking again the corresponding total morphisms as occurrences.

Free Strict Inverse Semigroups

A strict inverse semigroup can be constructed by means of the partially ordered set X = S/ J of its principal ideals, its J -classes which are the non-zero parts of Brandt semigroups, and partial homomorphisms between these partial semigroups. The free object in a variety of strict inverse semigroups is described in terms of these parameters. The associated partial order is realized as a set of symmetric and transitive relations on a certain set, and the corresponding Brandt semigroups as well as the partial homomorphisms can be obtained in a canonical way. As an application, a simple solution of the word problem in the free combinatorial strict inverse semigroup is presented.

The congruence lattice of a strict regular semigroup

The structure of a strict regular semigroup S is uniquely determined by the partially ordered set of its principal ideals S/ J , the J -classes of S which are the nonzero parts of completely 0-simple semigroups and certain partial homomorphisms between these partial groupoids. The congruences on S are described in terms of these parameters. Formulas for join and meet of a set of congruences are developed. As an application, necessary and sufficient conditions for a strict regular semigroup are obtained in order that its congruence lattice be contained in a variety of modular lattices.

Valuation Rings with Zero Divisors

Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis’ theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V.

Valuation rings with zero divisors

Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis’ theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V.

On partial homomorphisms of semigroups

On partial homomorphisms of semigroups

Extensions of completely 0-simple semigroups by completely 0-𝑠𝑖𝑚𝑝𝑙𝑒 𝑠𝑒𝑚𝑖𝑔𝑟𝑜𝑢𝑝𝑠.

Let S and T be disjoint semigroups, T having a zero element, O'. A semigroup (V, 0) is called an extension of S by T if it contains S as an ideal and if the Rees factor semigroup V/S is isomorphic to T. We shall say that V is determined by a partial homomorphism if there exists a partial homomorphism A -->: of T\O' into S such that A oB=AB if AB 5?O', A oB=AB if AB=O', A os=As, soA = sA, and s o t = st where s, t in S and the operations in S and T are denoted by juxtaposition. The purpose of this note is to give a necessary and sufficient condition that V be determined by a partial homomorphism when S is a completely 0-simple semigroup and T is a completely 0-simple semigroup. Since these partial homomorphisms are known mod group homomorphisms [1, p. 109, Theorem 3.14], our extensions may be given an explicit form. A corollary to our theorem will include an important theorem due to W. D. M unn [1, p. 143, Theorem 4.22]. Our result should have important applications to the study of finite semigroups and to semigroups with some finiteness condition. If S is any subset of a semigroup, 8(S) will denote the set of idempotents of S and S* will denote the set of nonzero elements of S. 6R, ?, 5D and 3C will denote Green's relations [1, p. 47]. If aES, Ra will denote the (R-class containing a. If e and f are idempotents and ef=fe = e, we say e is under f and write e <f. Basic definitions are given in [1]. Likewise references to the fundamental work of Clifford, Green, and Munn will be found in [1].

On Extensions of Semigroups Determined by Partial Homomorphisms

On Extensions of Semigroups Determined by Partial Homomorphisms

Partial homomorphic images of Brandt groupoids

The main purpose of the present note is to show (Theorem 2) that any regular D-class of any semigroup is a partial homomorphic image of a Brandt groupoid. It follows from this that a semigroup with zero is a partial homomorphic image of a Brandt semigroup if and only if it is regular and 0-bisimple. In the first section, an alternative formulation is given of the determination by H.-J. Hoehnke [1 ] of all partial homomorphisms of a Brandt groupoid into an arbitrary semigroup. This is first done (Theorem 1) for any completely 0-simple semigroup. The result is a straightforward generalization of Theorem 3.14 of [2], in which all partial homomorphisms of one completely 0-simple semigroup into another are determined. The present terminology is that of [2]; Hoehnke omits the adjective partial. Basic definitions given in [2] will not be repeated here; likewise, references to the fundamental work of Brandt, Rees, Green, and Munn can be found in [2].