Structure Of Semigroups Research Articles

Elaborating the word problem for free idempotent-generated semigroups over the full transformation monoid

With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $$\mathscr {E}$$ , one can construct the free idempotent-generated semigroup over $$\mathscr {E}$$ , $$\mathsf {IG}(\mathscr {E})$$ , which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $$\mathscr {E}$$ . Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, etc. In 2012, Gray and Ruškuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $$\mathscr {T}_n$$ , the full transformation monoid over an n-element set, might be worth pursuing. In 2019, together with Gould and Yang, the present author showed that the word problem for $$\mathsf {IG}(\mathscr {E}_{\mathscr {T}_n})$$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $$\mathscr {E}$$ , the algorithmic solution of the word problem revolves around the so-called vertex groups, which arise as certain subgroups of direct products of pairs of maximal subgroups of $$\mathsf {IG}(\mathscr {E})$$ . In this paper we determine these vertex groups for the case when $$\mathscr {E}$$ is the biorder of idempotents of $$\mathscr {T}_n$$ .

A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ›(R)

An ݑŸ-potent matrix in ݑ0ݑ›(R) is an ݑ› × ݑ› matrix satisfying ݐذݑŸ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ›(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) is known to be decomposable if rank (ݑ†) > ݑŸ − 1 for all ݑ† in S. Further, a semigroup S in ݑ0ݑ›(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑŸ-potent matrices in ݑ0ݑ›(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.

Brauer and partition diagram models for phylogenetic trees and forests.

We introduce a correspondence between phylogenetic trees and Brauer diagrams, inspired by links between binary trees and matchings described by Diaconis and Holmes (1998 Proc. Natl Acad. Sci. USA 95, 14 600–14 602. (doi:10.1073/pnas.95.25.14600)). This correspondence gives rise to a range of semigroup structures on the set of phylogenetic trees, and opens the prospect of many applications. We furthermore extend the Diaconis–Holmes correspondence from binary trees to non-binary trees and to forests, showing for instance that the set of all forests is in bijection with the set of partitions of finite sets.

On the structure of cancellative conjugation semigroups

On the structure of cancellative conjugation semigroups

The ( 2 , 3 ) -fuzzy set and its application in BCK-algebras and BCI-algebras

It is well-known that an intuitionistic fuzzy set is a generalization of a fuzzy set. Intuitionistic fuzzy sets deal with two types of fuzzy sets, namely membership function and non-membership function, under the condition that the sum of the membership degree and non-membership degree is less than or equal to \(1\). If the sum of membership degree and non-membership degree is greater than or equal to \(1\), the intuitionistic fuzzy set feels limited in its role. The Pythagorean fuzzy set that emerged to overcome these limitations is the generalization of the intuitionistic fuzzy set. As another form of generalization of the intuitionistic fuzzy set, the concept of the \((2,3)\)-fuzzy set is introduced in this article and several attributes are investigated. The semigroup structure is assigned to the collection of \((2,3)\)-fuzzy sets. The concepts of \((2,3)\)-fuzzy subalgebra for BCK/BCI-algebra and closed \((2,3)\)-fuzzy subalgebra for BCI-algebra are introduced and their properties are investigated. The relationship between the \((2,3)\)-fuzzy subalgebra and the degree function is discussed. A new \((2,3)\)-fuzzy subalgebra is generated using the given \((2,3)\)-fuzzy subalgebra. The union and intersection of \((2,3)\)-fuzzy subalgebras are addressed, and the characterization of the \((2,3)\)-fuzzy subalgebra using the \((2,3)\)-cutty set is addressed. Conditions for closing the \((2,3)\)-fuzzy subalgebra in the BCI-algebra are retrieved.

Higher Order Auxiliary Field Quantum Monte Carlo Methods

The auxiliary field quantum Monte Carlo (AFQMC) method has been a workhorse in the field of strongly correlated electrons for a long time and has found its most recent implementation in the ALF package (alf.physik.uni-wuerzburg.de). The utilization of the Trotter decomposition to decouple the interaction from the non-interacting Hamiltonian makes this method inherently second order in terms of the imaginary time slice. We show that due to the use of the Hubbard-Stratonovich transformation (HST) a semigroup structure on the time evolution is imposed that necessitates the introduction of a new family of complex-hermitian splitting methods for the purpose of reaching higher order. We will give examples of these new methods and study their efficiency, as well as perform comparisons with other established second and higher order methods in the realm of the AFQMC method.

Gaussian information bottleneck and the non-perturbative renormalization group

The renormalization group (RG) is a class of theoretical techniques used to explain the collective physics of interacting, many-body systems. It has been suggested that the RG formalism may be useful in finding and interpreting emergent low-dimensional structure in complex systems outside of the traditional physics context, such as in biology or computer science. In such contexts, one common dimensionality-reduction framework already in use is information bottleneck (IB), in which the goal is to compress an ‘input’ signal X while maximizing its mutual information with some stochastic ‘relevance’ variable Y. IB has been applied in the vertebrate and invertebrate processing systems to characterize optimal encoding of the future motion of the external world. Other recent work has shown that the RG scheme for the dimer model could be ‘discovered’ by a neural network attempting to solve an IB-like problem. This manuscript explores whether IB and any existing formulation of RG are formally equivalent. A class of soft-cutoff non-perturbative RG techniques are defined by families of non-deterministic coarsening maps, and hence can be formally mapped onto IB, and vice versa. For concreteness, this discussion is limited entirely to Gaussian statistics (GIB), for which IB has exact, closed-form solutions. Under this constraint, GIB has a semigroup structure, in which successive transformations remain IB-optimal. Further, the RG cutoff scheme associated with GIB can be identified. Our results suggest that IB can be used to impose a notion of ‘large scale’ structure, such as biological function, on an RG procedure.

Derivations on the algebras of the one-point compactification of affine semigroups

For any affine semigroup S, the set has a natural semigroup structure; additionally, if S is endowed with the discrete topology, then the semigroup can be studied as the one-point compactification of S. In this article, we study the derivations on semigroup algebra in relation to the derivations on semigroup algebra considering the metrizable topology on induced by the one-point compactification topology of

Semigroup Structures and Commutative Ideals of BCK-Algebras Based on Crossing Cubic Set Structures

First, semigroup structure is constructed by providing binary operations for the crossing cubic set structure. The concept of commutative crossing cubic ideal is introduced by applying crossing cubic set structure to commutative ideal in BCK-algebra, and several properties are investigated. The relationship between crossing cubic ideal and commutative crossing cubic ideal is discussed. An example to show that crossing cubic ideal is not commutative crossing cubic ideal is given, and then the conditions in which crossing cubic ideal can be commutative crossing cubic ideal are explored. Characterizations of commutative crossing cubic ideal are discussed, and the relationship between commutative crossing cubic ideal and crossing cubic level set is considered. An extension property of commutative crossing cubic ideal is established, and the translation of commutative crossing cubic ideal is studied. Conditions for the translation of crossing cubic set structure to be commutative crossing cubic ideal are provided, and its characterization is processed.

The concept of rhotrix type a semigroups

Several articles exist on rhotrices as associative rhotrix theory and non-associative rhotrix theory. In this paper, we describe the structure of rhotrix type semigroups. Ultimately, we extend the work of Mohammed et al on rhotrix semigroup.

Some Study of Semigroups of h-Bi-Ideals of Semirings.

Semigroups are generalizations of groups and rings. In the semigroup theory, there are certain kinds of band decompositions which are useful in the study of the structure of semigroups. This research will open up new horizons in the field of mathematics by aiming to use semigroup of h-bi-ideal of semiring with semilattice additive reduct. With the course of this research, it will prove that subsemigroup, the set of all right h-bi-ideals, and set of all left h-bi-ideals are bands for h-regular semiring. Moreover, it will be demonstrated that if semigroup of all h-bi-ideals (B(H), ∗) is semilattice, then H is h-Clifford. This research will also explore the classification of minimal h-bi-ideal.

A note on ensemble holography for rational tori

We study simple examples of ensemble-averaged holography in free compact boson CFTs with rational values of the radius squared. These well-known rational CFTs have an extended chiral algebra generated by three currents. We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants. In the simplest case, when the chiral algebra is primitive (in a sense we explain), the weights in this ensemble average are all equal. In the non-primitive case the ensemble weights are governed by a semigroup structure on the space of modular invariants.These observations can be viewed as evidence for a holographic duality between the ensemble of CFTs and an exotic gravity theory based on a compact U(1) × U(1) Chern-Simons action. In the bulk description, the extended chiral algebra arises from soliton sectors, and including these in the path integral on thermal AdS3 leads to the vacuum character of the chiral algebra. We also comment on wormhole-like contributions to the multi-boundary path integral.

The lattice and semigroup structure of multipermutations

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an [Formula: see text]-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in [Formula: see text] or to be [Formula: see text]-complete. We go on to study the monoid of all multipermutations on an [Formula: see text]-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on [Formula: see text].

On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups

We study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain 𝔒 obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of “upper semicontinuous functions” whose domain is the maximal spectrum of 𝔒 equipped with the inverse topology introduced by Hochster, and whose codomain is a certain totally ordered monoid containing ℝ. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of 𝔒. This result can be regarded as a generalization of the theory of prime ideal factorization for Dedekind domains. By using such isomorphism, we study the Galois-monoid structure of the ideal class semigroup of 𝔒.

Locally Roelcke precompact Polish groups

A Polish group is locally Roelcke precompact if there is a neighborhood of the identity element that is totally bounded in the Roelcke (or lower) group uniformity. These form a subclass of the locally bounded groups, while generalizing the Roelcke precompact and locally compact Polish groups. We characterize these groups in terms of their geometric structure as those locally bounded groups whose coarsely bounded sets are all Roelcke precompact, and in terms of their uniform structure as those groups whose completions in the Roelcke uniformity are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup.

Partial actions on reductive Lie algebras

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center.

Reconstructing étale groupoids from semigroups

AbstractWe unify various étale groupoid reconstruction theorems such as the following:•Kumjian and Renault’s reconstruction from a groupoid C*-algebra;•Exel’s reconstruction from an ample inverse semigroup;•Steinberg’s reconstruction from a groupoid ring;•Choi, Gardella and Thiel’s reconstruction from a groupoidLp{L^{p}}-algebra.We do this by working with certainbumpysemigroupsSof functions defined on an étale groupoidG. The semigroup structure ofStogether with the diagonal subsemigroupDthen yields a natural domination relation≺{\prec}onS. The groupoid of≺{\prec}-ultrafilters is then isomorphic to the original groupoidG.

Dilation theory for right LCM semigroup dynamical systems

This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C⁎-algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representation of a right LCM system can be dilated to an isometric representation of the boundary quotient.

A combinatorial approach to the structure of locally inverse semigroups

This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the role that Cayley graphs have for group presentations or that Sch\"utzenberger graphs have for inverse monoid presentations. However, our graphs have considerable differences with the latter two, even though locally inverse semigroups generalize both groups and inverse semigroups. For example, the graphs introduced here are not `inverse word graphs'. Instead, they are bipartite graphs with both oriented and non-oriented edges, and with labels on the oriented edges only. A byproduct of the theory developed here is the introduction of a graphical method for dealing with general locally inverse semigroups. These graphs are able to describe, for a locally inverse semigroup given by a presentation, many of the usual concepts used to study the structure of semigroups, such as the idempotents, the inverses of an element, the Green's relations, and the natural partial order. Finally, the paper ends characterizing the semigroups belonging to some usual subclasses of locally inverse semigroups in terms of properties on these graphs.

Ordered abundant semigroups with adequate transversals

Abundant semigroups containing multiplicative ample transversals are introduced by the author. Guo establishes the structure of abundant semigroups with multiplicative adequate transversals. As any ample semigroup is adequate, is Guo’s structure an extension of that introduced by the author? After answering this question positively and providing a natural order for the multiplicative ample transversal case, an order analogue of Guo’s structure is also demonstrated. Since left normal bands and right normal bands, each with a skeleton, play a vital role in establishing the structure of the classes of abundant semigroups under consideration, some attention has been paid to the structure of such bands.