NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES

2008 ◽  
Vol 11 (03) ◽  
pp. 351-375 ◽  
Author(s):  
ROMUALD LENCZEWSKI ◽  
RAFAŁ SAŁAPATA
Keyword(s):  
Partial Order ◽  
Fock Space ◽  
Free Probability ◽  
Simple Relation ◽  
Poisson Processes ◽  
Natural Partial Order ◽  
Brownian Motions ◽  
Continuous Interpolation ◽  
Monotone Probability ◽  
Two Parameter

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

scholarly journals Homomorphisms and congruences on ωα-bisimple semigroups

1973 ◽  
Vol 15 (4) ◽  
pp. 441-460 ◽  
Author(s):  
J. W. Hogan
Keyword(s):  
Partial Order ◽  
Ordered Set ◽  
Order Relation ◽  
Natural Partial Order ◽  
Partial Order Relation

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

scholarly journals Primitive idempotent measures on compact semitopological semigroups

1972 ◽  
Vol 13 (4) ◽  
pp. 451-455 ◽  
Author(s):  
Stephen T. L. Choy
Keyword(s):  
Partial Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Zero Element ◽  
Primitive Idempotent ◽  
Natural Partial Order ◽  
Partially Ordered

For a semigroup S let I(S) be the set of idempotents in S. A natural partial order of I(S) is defined by e ≦ f if ef = fe = e. An element e in I(S) is called a primitive idempotent if e is a minimal non-zero element of the partially ordered set (I(S), ≦). It is easy to see that an idempotent e in S is primitive if and only if, for any idempotent f in S, f = ef = fe implies f = e or f is the zero element of S. One may also easily verify that an idempotent e is primitive if and only if the only idempotents in eSe are e and the zero element. We let П(S) denote the set of primitive idempotent in S.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

scholarly journals Interval-type theorems concerning means

2018 ◽  
Vol 17 (1) ◽  
pp. 37-43
Author(s):  
Paweł Pasteczka
Keyword(s):  
Partial Order ◽  
Natural Isomorphism ◽  
Natural Partial Order ◽  
Real Numbers ◽  
Power Means ◽  
Interval Type ◽  
Type Sets

Abstract Each family ℳ of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x. In this setting we can introduce the notion of interval-type set (a subset ℐ ⊂ℳ such that whenever M ≤ P ≤ N for some M, N ∈ℐ and P ∈ℳ then P ∈ℐ). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. In the present paper we consider this property for Gini means and Hardy means. Moreover, some results concerning L∞ metric among (abstract) means will be obtained.

Cumulative sum control charts for monitoring geometrically inflated Poisson processes: An application to infectious disease counts data

2016 ◽  
Vol 27 (2) ◽  
pp. 622-641 ◽  
Author(s):  
Athanasios C Rakitzis ◽  
Philippe Castagliola ◽  
Petros E Maravelakis
Keyword(s):  
Control Charts ◽  
Statistical Design ◽  
Poisson Processes ◽  
Cumulative Sum ◽  
The Usa ◽  
Control Situations ◽  
Zero Values ◽  
Cumulative Sum Control ◽  
Two Parameter ◽  
Excessive Number

In this work, we study upper-sided cumulative sum control charts that are suitable for monitoring geometrically inflated Poisson processes. We assume that a process is properly described by a two-parameter extension of the zero-inflated Poisson distribution, which can be used for modeling count data with an excessive number of zero and non-zero values. Two different upper-sided cumulative sum-type schemes are considered, both suitable for the detection of increasing shifts in the average of the process. Aspects of their statistical design are discussed and their performance is compared under various out-of-control situations. Changes in both parameters of the process are considered. Finally, the monitoring of the monthly cases of poliomyelitis in the USA is given as an illustrative example.

scholarly journals On bisimple semigroups generated by a finite number of idempotents

1982 ◽  
Vol 33 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Karl Byleen
Keyword(s):  
Finite Number ◽  
Partial Order ◽  
Natural Partial Order ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractNon-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

scholarly journals Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups

1981 ◽  
Vol 31 (3) ◽  
pp. 325-336 ◽  
Author(s):  
D. B. Mcalister
Keyword(s):  
Partial Order ◽  
Local Structure ◽  
Structure Theorem ◽  
Main Tool ◽  
Natural Partial Order ◽  
Ordered Semigroup ◽  
Ordered Group ◽  
Partially Ordered ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups

AbstractA partially ordered semigroup S is said to be a Dubreil-Jacotin semigroup if there is an isotone homomorphism θ of S onto a partially ordered group such that {} has a greatest member. In this paper we investigate the structure of regular Dubreil-Jacotin semigroups in which the imposed partial order extends the natural partial order on the idempotents. The main tool used is a local structure theorem which is introduced in Section 2. This local structure theorem applies to many other contexts as well.

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