Semigroup Homomorphisms Research Articles

THE HOMOTOPY THEORY OF INVERSE SEMIGROUPS

We show that abstract homotopy theory can be used to define a suitable notion of homotopy equivalence for inverse semigroups. As an application of our theory, we prove a theorem for inverse semigroup homomorphisms which is the exact counterpart of the well-known result in topology which states that every continuous function can be factorized into a homotopy equivalence followed by a fibration. We show that this factorization is isomorphic to the one constructed by Steinberg in his "Fibration Theorem", originally proved using a generalization of Tilson's derived category.

Semigroup homomorphisms and fuzzy automata

A generalized Ω-fuzzy automaton over a complete residuated lattice Ω and a monoid (M,*) and with a set S of states is introduced as a monoid homomorphism F:(M,*)→(?,∘), where (?,∘) is a monoid of Ω-fuzzy sets in a set S×S. An extension principle depending of proper filters Φ in Ω is introduced which then enables to introduce morphisms between generalized Ω-fuzzy automata and to introduce the category ℱΦ of these automata. It is proved that categories of classical fuzzy automata, non-deterministic automata and some other systems are equivalent to subcategories of ℱΦ for a suitable filter Φ.

More on matrix semigroup homomorphisms

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all 2-by-2 complex matrices to the semigroup of n-by- n complex matrices. If such a homomorphism maps a cyclic unipotent to a cyclic unipotent, it is the composition of a symmetric power, a field homomorphism used entrywise, and a matrix conjugation. In the case n=4 we characterise all non-degenerate irreducible homomorphisms.

FACTORIZATION THEOREMS FOR MORPHISMS OF ORDERED GROUPOIDS AND INVERSE SEMIGROUPS

AbstractAdapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

Inverse semigroup homomorphisms via partial group actions

This papar constructs all homomorphisms of inverse semigroups which factor through anE-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domainIof βE-unitary; moreover, theP-representation ofIis explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism isE-unitary. Stronger results are obtained for the case ofF-inverse monoids.Special cases of our results include theP-theorem and the factorisation theorem for homomorphisms fromE-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence ofE-unitary covers over a group, as well as a generalisation toF-inverse covers, allowing a quick proof that every inverse monoid has anF-inverse cover.

On the dual of complex Ol'shanski\uı semigroups

LetG be a connected Lie group which sits in its universal complexification GC. If g denotes the Lie algebra of G and W ⊆ g is a non-empty open Ad(G)-invariant convex cone containing no affine lines, then a complex Ol’shanskii semigroup is defined by S = G exp(iW ) ⊆ GC. One knows that S is an open subsemigroup ofGC with holomorphic multiplication, and moreover S is invariant under the antiholomorphic involution g → g∗ = g−1, where g indicates complex conjugation in GC. In particular, (S, ∗) is an involutive semigroup. If H is a Hilbert space and B(H) denotes the bounded operators on it, then a holomorphic representation (π,H) of S is a holomorphic semigroup homomorphism π:S → B(H)with total image andwhich satisfies π(s∗) = π(s)∗ for all s ∈ S. A mapping α:S → [0,∞[ satisfying α(s∗) = α(s) and α(st) ≤ α(s)α(t) for all s, t ∈ S is called an absolute value of S. We call a holomorphic representation α-bounded for some absolute value α if ‖π(s)‖ ≤ α(s) holds for s ∈ S. The α-bounded holomorphic representations of S can be modelled via a certain C∗-algebra C∗ h(S, α) (cf. Definition I.3, Lemma II.6), i.e., there is a natural correspondence between α-bounded holomorphic representations of S and non-degenerate representations of C∗ h(S, α). An important result of K.-H. Neeb asserts that these C∗-algebras C∗ h(S, α) are CCR (cf. [Ne99, Ch. XI]). If we denote by Ŝα the set of equivalence classes of irreducible α-bounded representations of S, we therefore obtain a bijection

A new efficient computational algorithm for bit reversal mapping

In this correspondence, we present a new bit reversal algorithm that outperforms the existing ones. The bit reversal technique is involved in the fast fourier transform (FFT) technique, which is widely used in computer-based numerical techniques for solving numerous problems. The new approach for computing the bit reversal is based upon a pseudo semi-group homomorphism property. The surprising thing is that this property is almost trivial to prove, but at the same time, it also leads to a very efficient algorithm, which we believe to be the best with only O(N) operations and optimal constant, i.e. unity.

Matrix semigroup homomorphisms from dimension two to three

We characterise all non-degenerate homomorphisms from the multiplicative semigroup of all 2×2 matrices over an arbitrary field to the semigroup of 3×3 matrices over the same field. In the case of a field of real numbers every irreducible non-degenerate homomorphism is a conjugation of the symmetric square.

Strong continuity of semigroup homomorphisms

Let J be an abelian topological semigroup and C a subset of a Banach space X. Let L(X) be the space of bounded linear operators on X and Lip(C) the space of Lipschitz functions ⨍: C → C. We exhibit a large class of semigroups J for which every weakly cont

Semigroup properties of factors in the polar decomposition or the operator De-Moivre formula

We give necessary and sufficient conditions under which the factors in the polar decomposition of a semigroup homomorphism are themselves semigroup homomorphisms.

Homomorphisms of implicative semigroups

Implicative semigroups and Brouwerian semigroups were studied by W. C. Nemitz and T. S. Blyth, respectively. In this paper, following the ideas of Nemitz and Blyth, we introduce the notion of negatively partially ordered implicative semigroups and studied the homomorphisms between these semigroups. Some results of Nemitz on implicative semilattices are generalized and amplified to implicative semigroups.

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

Homomorphisms of sandwich semigroups and sandwich near-rings

Performance estimates for applications: An algebraic framework

In the approach described in this paper the performance of an application is described via a network of its building blocks. If each of the building blocks is assigned a value in the ‘performance algebra’, which turns out to be an ordered semigroup, the total performance can be obtained by applying a semigroup homomorphism. Data movement, arithmetic operations and delays (such as dispatching overhead) and their interactions can be algebraically modelled. The method can be applied to whole applications but also to operations close to the instruction level. Hockney's laws and Amdahl's law can be retrieved as well as the law of the harmonic mean. In addition, a number of inequalities for algebraic constructs are given and the relationship between a ‘machine independent’ and a ‘machine dependent’ application algebra is studied. We formulate an ‘abstract benchmarking problem’ and discuss a number of applications.

A closure property of regular languages

Given a semigroup homomorphism θ: X ∗→ P ( Y ∗) where P ( Y ∗) is the powerset of Y ∗, we show that the inverse image Kθ -1 of a regular set K ⊆ Y ∗ is regular, too. This theorem has several consequences generalizing previously known results.

Completions for partially ordered semigroups

A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called “subset systems”. Related facts are derived for conditional completions.

Modified Whyburn semigroups

Let f : X → Y be a continuous semigroup homomorphism. Conditions are given which will ensure that the semigroup X ∪ Y is a topological semigroup, when the modified Whyburn topology is placed on X ∪ Y.

Cumulated social roles: The duality of persons and their algebras

The study of social roles from the perspectives of individual actors, and the relation of graph homomorphisms to semigroup homomorphisms, have been the two most prominent topics to emerge from the recent resurgence of progress made on the algebraic analysis of social networks. Through our central construction, the cumulated person hierarchy, we present a framework for elaborating and extending these two lines of research. We focus on each actor in turn as ego, and we articulate what we believe to be the fundamental duality of persons and their algebras. We derive graph and semigroup homomorphisms for three algebras containing 81, 43, and 93 elements, respectively. Throughout, our discussion of theoretical issues is oriented toward an empirical application to the Padgett data set on conspiracy and faction in Renaissance Florence.

The stability of multiplicative semigroup homomorphisms to real normed algebras, I

The stability of multiplicative semigroup homomorphisms to real normed algebras, I

Group relationships and homomorphisms of Boolean matrix semigroups

The image matrix obtained from a blocked Boolean matrix gives a homomorphism provided that the product of any two nonzero blocks is nonzero. All maximal nonzero Boolean matrix semigroups are found. Those invariant under conjugations by a permutation matrix have the form G i = { A: the sum of any ith rows of A (or all rows) has at least i ones (or all ones)}.