Structure Of Semigroups Research Articles

Structure of block quantum dynamical semigroups and their product systems

Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

Combinatorial characterization of pseudometrics

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y \to X$ such that $\Psi(x, y) = f(\Phi(g(x), g(y)))$ for all $x$, $y \in Y$. It is shown that the semigroups of binary relations generated by sets $\{\Phi^{-1}(a) \colon a \in \Phi(X^{2})\}$ and $\{\Psi^{-1}(b) \colon b \in \Psi(Y^{2})\}$ are isomorphic for combinatorially similar $\Phi$ and $\Psi$. The necessary and sufficient conditions under which a given mapping is combinatorially similar to a pseudometric, or strongly rigid pseudometric, or discrete pseudometric are found. The algebraic structure of semigroups generated by $\{d^{-1}(r) \colon r \in d(X^{2})\}$ is completely described for nondiscrete, strongly rigid pseudometrics and, also, for discrete pseudometrics $d \colon X^{2} \to \mathbb{R}$.

Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism

Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$ has finitely many connected components; and every amplified endomorphism is int-amplified.

Groupoids on a skew lattice of objects

Motivated by some alternatives to the classical logical model of boolean algebra, this paper deals with algebraic structures which extend skew lattices by locally invertible elements. Following the meme of the Ehresmann-Schein-Nambooripad theorem, we consider a groupoid (small category of isomorphisms) in which the set of objects carries the structure of a skew lattice. The objects act on the morphisms by left and right restriction and extension mappings of the morphisms, imitating those of an inductive groupoid. Conditions are placed on the actions, from which pseudoproducts may be defined. This gives an algebra of signature (2, 2, 1), in which each binary operation has the structure of an orthodox semigroup. In the reverse direction, a groupoid of the kind described may be reconstructed from the algebra.

Multiplicative factorization in numerical semigroups

A numerical semigroup is a nonempty additively closed subset [Formula: see text] with [Formula: see text]. The arithmetic, that is the additive factorization properties, of numerical semigroups has been well studied. Their multiplicative properties, on the other hand, have received little, if any, attention. If [Formula: see text] or [Formula: see text], then multiplicative factorization (as products of primes) in [Formula: see text] is unique. However, if there is [Formula: see text] with [Formula: see text], then multiplicative factorization in [Formula: see text] is no longer unique. The purpose of this paper is to introduce this previously unstudied structure of numerical semigroups. Specifically, we classify the irreducible elements and provide a description of how non-unique multiplicative factorization can be in numerical semigroups. In addition, we show that multiplicative numerical semigroups belong to the class of [Formula: see text]-monoids, but yet are not Krull.

A Black\u2013Scholes inequality: applications and generalisations

The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.

A new semigroup obtained via known ones

The goal of this paper is to establish a new class of semigroups based on both Rees matrix and completely [Formula: see text]-simple semigroups. We further present some fundamental properties and finiteness conditions for this new semigroup structure.

Cross-connection structure of concordant semigroups

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalized Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalization of the cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

The graphic structure of a numerical semigroup

Motivated by the way numerical semigroups arise from primitive digraphs, we initiate the study of the graphic structure of a numerical semigroup. We lay some general groundwork for determining this structure and apply our results to a minimization problem.

Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure

We study measures on a real separable Hilbert space E that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of E that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $${\mathcal H}$$ of complex-valued functions on E that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on E that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $${\mathcal H}$$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.

Structural Folding and Multi-Highest-Weight Subcrystals of $B(\infty )$

We introduce a procedure to fold the structure of a crystal B of simply-laced Cartan type \({\mathscr{C}}\) by the action of an automorphism σ. This produces a crystal Bσ for the folded Langlands dual datum \({\mathscr{C}}^{\sigma \vee }\) which properly contains the well-studied \({\mathscr{C}}^{\sigma \vee }\) crystal of σ-invariant points. Our construction preserves normality and the Weyl group action, and is compatible with Kashiwara’s tensor product rule. Combinatorial properties of \(B(\infty )_{\sigma }\) reflect the structure of a subalgebra of \(U_{q}^{-}({\mathscr{C}})\), which is naturally a module over the graded-σ-fixed-point subalgebra of \(U_{q}^{-}({\mathscr{C}})\) via Berenstein and Greenstein’s quantum folding procedure. We find that \(B(\infty )_{\sigma }\) is generated by a set of highest-weight elements over the monoid of root operators. Through the Kashiwara-Nakashima-Zelevinsky polyhedral realization, the highest-weight set identifies with a commutative monoid which admits a Hilbert basis in finite type. A subset of the Weyl group called the balanced parabolic quotient is in one-to-one correspondence with the Hilbert basis for the pair \(\left ({\mathscr{C}}, {\mathscr{C}}^{\sigma \vee } \right ) = \left (D_{r}, C_{r-1} \right )\), and identifies with a proper subset of the Hilbert basis in other finite types. We obtain an explicit combinatorial description of the highest-weight set of \(B(\infty )_{\sigma }\) by establishing a connection between the action of root operators on \(B(\infty )\) and the semigroup structure in the polyhedral realization.

Structure of Strict Abundant Semigroups

We introduce a new class of semigroups called strict abundant semigroups, which are concordant semigroups and subdirect products of completely [Formula: see text]-simple abundant semigroups and completely 0-[Formula: see text]-simple primitive abundant semigroups. A general construction and a tree structure of such semigroups are established. Consequently, the corresponding structure theorems for strict regular semigroups given by Auinger in 1992 and by Grillet in 1995 are generalized and extended. Finally, an example of strict abundant semigroups is also given.

KONDISI MINIMAL IDEAL KIRI TERURUT PADA SEMIGRUP TERNER TERURUT PARSIAL

Ternary semigroups 𝑇 is obtained from a nonempty set 𝑇 that given a mapping with a multiplication operation ternary that satisfied closed and associative properties. So, generally a ternary semigroup is an abstraction of a semigroup structure. Meanwhile, partially ordered ternary semigroups 𝑇 is an ordered semigroup 𝑇 that satisfies the properties for each 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑇 if 𝑎 ≤ 𝑏 then (𝑎𝑐𝑑) ≤ (𝑏𝑐𝑑) and (𝑑𝑐𝑎) ≤ (𝑑𝑐𝑏). In a ternary semigroups there is also concept of left ideals. This study was conducted to examine the characteristics of ordered left ideals on partially ordered ternary semigroups. Furthermore, it will be discussed about the characteristics of minimal ordered left ideals on partially ordered semigroups.Keywords : Ternary Semigroups, Ordered Ternary Semigroups, Left Ideals, Ordered Left Ideals, Minimal of Ordered Left Ideals.

SEMIGROUPS GENERATED BY PARTITIONS

Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of $X^{2}$ and, respectively, by the finest symmetric partition of $X^{2}$ are described.

An alternative look at the structure of graph inverse semigroups

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

A New Proof of the Protasov-Voynov Theorem on Semigroups of Nonnegative Matrices

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.

On the lattice of biorder ideals of regular rings

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup [Formula: see text] of a regular ring [Formula: see text] by studying the idempotents [Formula: see text] of the regular ring and show that the principal biorder ideals of the regular ring [Formula: see text] form a complemented modular lattice and certain properties of this lattice are studied.

Quantum semigroups from synchronous games

We show that the C*-algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game. In particular situations (e.g., for graph endomorphism games), these quantum families have a natural quantum semigroup structure, and if the condition of preservation of a natural state is added, they are in fact compact quantum groups.

On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers

For a commutative integral domain, the class semigroup and the class group are defined as the quotient of the semigroup of fractional ideals and the group of invertible ideals by the group of principal ideals, respectively. Let $p$ be a prime number. In algebraic number theory, especially in Iwasawa theory, the class group of the ring of integers $\mathcal{O} $ of the cyclotomic $\mathbb {Z}_{p}$-extension of the rational numbers has been studied for a long time. However, the class semigroup of $\mathcal{O} $ is not well known. We are interested in the structure of the class semigroup of $\mathcal{O} $. In order to study it, we focus on the structure of the complement set of the class group in the class semigroup of $\mathcal{O} $. In this paper, we prove that the complement set is a group and determine its structure.

Hypergroup deformations of semigroups

We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set $\mathbb{Z}_+$ of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from "max" semigroups or general commutative semigroups via hypergroup deformation on idempotents.