Structure Of Semigroups Research Articles

A golden iterated map number system: Results and conjectures

We construct a number system for representing numbers in [0, 1] that is based on iterations of an asymmetric tent map that incorporates the golden ratio. It is already known that in this system a number has a periodic representation if and only if it lies in [Formula: see text]. We investigate other aspects of this system such as non-uniquely representable numbers, an inherent semigroup structure, connections with Wythoff's game, related sequences of rational functions, and its connection with an iterative scheme reminiscent of paper folding analysis. Many details of the above-mentioned connections are conjectural.

On Some Monogenic Semigroups with an Associate Subgroup

Let S be any semigroup and a, s ∈ S. If a = asa, then s is an associate of a. A subgroup G of S is an associate subgroup of S if every a ∈ S has a unique associate a* in G. It turns out that G = H z for some idempotent z, the zenith of S. The mapping a → a* is a unary operation on S. We say that S is monogenic if S is generated, as a unary semigroup, by a single element. We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open.

The structure of certain F-inverse semigroups

An inverse semigroup is an F-inverse semigroup if each class modulo the minimum group congruence has a maximum element. This paper studies a class of F-inverse semi-groups in which all these maximum elements form a subgroup. After obtaining some characterizations of such semigroup, the structure theorem is obtained by the semidirect product of a semilattice and a group.

Infinite-dimensional -adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings

We construct -adic analogues of operator colligations and their characteristic functions. Consider a -adic group , a subgroup of and a subgroup which is diagonally embedded in . We show that the space of double cosets admits the structure of a semigroup and acts naturally on the space of -fixed vectors of any unitary representation of . With each double coset we associate a `characteristic function' that sends a certain Bruhat– Tits building to another building (the buildings are finite- dimensional) in such a way that the image of the distinguished boundary lies in the distinguished boundary. The second building admits the structure of a (Nazarov) semigroup, and the product in corresponds to the pointwise product of characteristic functions.

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T-graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T-graded algebra is unital, or T is a cancellative semigroup, then the T-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.

Combinatorial Properties of Entire Semigroups of Nonnegative Matrices

Generalizations of the Protasov–Voynov theorem on the structure of irreducible semigroups of nonnegative matrices free of zero rows and columns are obtained. The theorem is extended to semigroups that are allowed to be reducible and to matrices that may have zero columns. The main results concern the semigroups called entire. In the definitions and proofs, only combinatorial properties of nonnegative matrices are exploited. Bibliography: 7 titles

Dynamic RWX ACM Model Optimizing The Risk on Real Time Unix File System

The preventive control is one of the well advance controls for recent security for protection of data and services from the uncertainty. Because, increasing the importance of business, communication technologies and growing the external risk is a very common phenomenon now-a-days. The system security risks put forward to the management focus on IT infrastructure (OS). The top management has to decide whether to accept expected losses or to invest into technical security mechanisms in order to minimize the frequency of attacks, thefts as well as uncertainty. This work contributes to the development of an optimization model that aims to determine the optimal cost to be invested into security mechanisms deciding on the measure component of UFS attribute. Our model should be design in such way, the Read, Write & Execute automatically Protected, Detected and Corrected on RTOS. We have to optimize the system attacks and down time by implementing RWX ACM mechanism based on semi-group structure, mean while improving the throughput of the Business, Resources & Technology. DOI: http://dx.doi.org/10.11591/telkomnika.v13i2.7059 Full Text: PDF

On the structure of numerical sparse semigroups and applications to Weierstrass points

In this work, we are concerned with the structure of sparse semigroups and some applications of them to Weierstrass points. We manage to describe, classify and find an upper bound for the genus of sparse semigroups. We also study the realization of some sparse semigroups as Weierstrass semigroups. The smoothness property of monomial curves associated to (hyper)ordinary semigroups presented by Pinkham and Rim–Vitulli, and the results on double covering of curves by Torres are crucial in this.

On Salem numbers, expansive polynomials and Stieltjes continued fractions

A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $\beta$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions over a Salem polynomial having simple roots, not cancelling at $\pm 1$, coming from monic expansive polynomials of constant term equal to their Mahler measure, has a semigroup structure. The sets of corresponding generalized Garsia numbers inherit this semi-group structure.

Some model theory of SL(2,R)

We study the action of G = SL(2;R), viewed as a denable in the structure M = (R; +; ), on its type space SG(M). We identify a minimal closed G-ow I, and an idempotent r2 I (with respect to the Ellis semigroup structure on SG(M)). We also show that the \ideal group (r I; ) is nontrivial (in fact it will be the with 2 elements), yielding a negative answer to a question of Newelski.

Pseudocompact Primitive Topological Inverse Semigroups

In the paper we study pseudocompact primitive topological inverse semigroups. We describe the structure of pseudocompact primitive topological inverse semigroups and show that the Tychonoff product of a family of pseudocompact primitive topological inverse semigroups is a pseudocompact topological space. Also we prove that the Stone – Cech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup.

On the structure of semigroups on Lp with a bounded H∞ -calculus

We show that a bounded analytic semigroup on a reflexive L p -space has a bounded H ∞ ( Σ θ ) -calculus for some θ < π / 2 if and only if the semigroup can be obtained, after restricting to invariant subspaces, factorizing through invariant subspaces and similarity transforms, from a bounded analytic semigroup on some bigger L p -space which is positive and contractive on the real line.

Continuous semigroup structures on $$\mathbb {R}$$ R , cancellative semigroups and bands

Up to isomorphism there is only one continuous group operation on \(\mathbb {R}\), as Aczel (Bull Soc Math Fr 76: 59–64, 1949) showed in 1949. In this paper we show that there are exactly three distinct continuous (weakly) cancellative semigroup structures on \(\mathbb {R}\) modulo isomorphism. On the other hand, there are many continuous non-cancellative semigroup structures on \(\mathbb {R}\). We classify all ordered continuous bands (idempotent semigroups) on \(\mathbb {R}\). There are exactly eight (five) distinct ordered continuous band structures on \(\mathbb {R}\) modulo isomorphism (and anti-isomorphism).

Applications of Homomorphism on the Structure of Semigroups

By utilizing homomorphisms and -strong semilattice of semigroups, we show that the Green (*,~)-relation H*,~ is a regular band congruence on a r-ample semigroup if and only if it is a G-strong semilattice of completely J*,~-simple semigroups. The result generalizes Petrich’s result on completely regular semigroups with Green’s relation H a normal band congruence or a regular band congruence from the round of regular semigroups to the round of r-ample semigroups.

On compact quantum semigroup QS red

We study some properties of a reduced semigroup C*-algebra of a semigroup S. For the semigroup C*-algebra generated by the deformation of the algebra of continuous functions on a compact abelian group we obtain a structure of a compact quantum semigroup. We also consider morphisms of constructed compact quantum semigroups.

THE SEMIGROUPS B2AND B0ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

The five-element Brandt semigroup B2and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅,-1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1and retaining the induced operations x ↦ x+= xx-1and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids".These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0and B2.

THE PRE-LIE OPERAD AS A DEFORMATION OF NAP

We introduce current-preserving operads, which are colored operads together with an additive semigroup structure on the set G of colors. This is the natural framework in which types of G-graded algebras can be dealt with. As an example of application, we deform the NAP operad into the pre-Lie operad in this context, where G is the additive semigroup of positive integers. More precisely, we give a family [Formula: see text] of current-preserving operads, depending on a scalar parameter λ, such that [Formula: see text], respectively, [Formula: see text], is the current-preserving operad governing graded NAP (respectively, pre-Lie) algebras.

Topological dynamics and definable groups

AbstractWe give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G(M) on its “external type space” SG.ext(M), can explain, account for, or give rise to, the quotient G/G00, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or f sg) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of G(M) for measure-stable groups. We also point out that for G definably amenable (in a NIP theory) G/G00 can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global f-generic types.

A DESCRIPTION OF A CLASS OF FINITE SEMIGROUPS THAT ARE NEAR TO BEING MAL'CEV NILPOTENT

In this paper we continue the investigation on the algebraic structure of a finite semigroup S that is determined by its associated upper non-nilpotent graph [Formula: see text]. The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Mal'cev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of pseudo-nilpotent semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is pseudo-nilpotent if and only if it is nilpotent. Our main result is a description of pseudo-nilpotent finite semigroups S in terms of their associated graph [Formula: see text]. In particular, S has a largest nilpotent ideal, say K, and S/K is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements.

Homomorphisms and congruences on regular semigroups with associate inverse subsemigroups

An associate inverse subsemigroup of a regular semigroup S is a subsemigroup T of S containing a least associate x* of each x ∈ S, in relation to the natural partial order ≤. In [1] the authors describe the structure of regular semigroups with an associate inverse subsemigroup, satisfying two natural conditions. In this paper we describe all *-homomorphisms and all *-congruences on such semigroups.