Noncommutative inclusion–exclusion, representations of left regular bands and the Tsetlin library

2020 ◽  
pp. 1-26
Author(s):  
Guy Blachar ◽  
Louis H. Rowen ◽  
Uzi Vishne
Keyword(s):  
Regular Band ◽  
Left Regular ◽  
Free Left ◽  
Left Regular Band ◽  
The Absolute ◽  
Left Regular Bands

We find a semigroup [Formula: see text], whose category of partial representations contains the representation category [Formula: see text] of the free left regular band [Formula: see text]. We use this to construct a resolution for the absolute kernel of a representation of [Formula: see text], for which the kernel [Formula: see text] of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of [Formula: see text] to the number of derangements of order [Formula: see text].

scholarly journals Left regular bands of groups of left quotients

1991 ◽  
Vol 33 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Abdulsalam El-Qallali
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Inverse Semigroups ◽  
Left Inverse ◽  
Regular Band ◽  
Left Regular ◽  
Left Regular Band ◽  
Left Regular Bands

In this paper we characterize semigroups S which have a semigroup Q of left quotients, where Q is an ℛ-unipotent semigroup which is a band of groups. Recall that an ℛ-unipotent (or left inverse) semigroup S is one in which every ℛ-class contains a unique idempotent. It is well-known that any ℛ-unipotent semigroup 5 is a regular semigroup in which the set of idempotents is a left regular band in that efe = ef for any idempotents e, fin S. ℛ-unipotent semigroups were studied by several authors, see for example [1] and [13].Bailes [1]characterized ℛ-unipotent semigroups which are bands of groups. This characterization extended the structure of inverse semigroups which are semilattices of groups. Recently, Gould studied in [7]the semigroup S which has a semigroup Q of left quotients where Q is an inverse semigroup which is a semilattice of groups.

scholarly journals Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

2021 ◽  
Vol 274 (1345) ◽  
Author(s):  
Stuart Margolis ◽  
Franco Saliola ◽  
Benjamin Steinberg
Keyword(s):  
Representation Theory ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Oriented Matroid ◽  
Regular Band ◽  
Left Regular ◽  
The Face ◽  
Left Regular Band ◽  
Poset Topology ◽  
Left Regular Bands

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Finite submonoids of free left regular bands with unit

2021 ◽  
Vol 59 (6) ◽  
pp. 680-701
Author(s):  
A. N. Shevlyakov
Keyword(s):  
Left Regular ◽  
Free Left ◽  
Left Regular Bands

GENERALIZATION OF F-INVERSE SEMIGROUPS BY USING MCALISTER'S APPROACH

2008 ◽  
Vol 01 (04) ◽  
pp. 535-553
Author(s):  
Xiaojiang Guo ◽  
Xiangfei Ni ◽  
K. P. Shum
Keyword(s):  
Direct Product ◽  
Inverse Semigroups ◽  
New Method ◽  
Regular Band ◽  
Cancellative Monoid ◽  
Left Regular ◽  
Left Regular Band

We generalize the F-inverse semigroups within the class of lpp-semigroups by using McAlister's approach and FGC-systems. Consider a left GC - lpp monoid M. If M is lpp, then M is called a left F-pseudo group and for brevity, we call the semi-direct product of a left regular band and a cancellative monoid a twisted left cryptic group. In this paper, the structures of left F-pseudo groups are investigated. It is shown that a left F-pseudo group whose minimum right cancellative monoid congruence is cancellative can be embedded into a twisted left cryptic group. This result generalizes a number of known results in F-inverse semigroups previously given by C. C. Edwards, R. B. McFadden, L. O'Carrol, X. J. Guo and others. In particular, a new method constructing F -right inverse semigroups is provided.

Finite Submonoids of Free Left Regular Bands with Unit

2021 ◽  
Vol 59 (6) ◽  
pp. 456-470
Author(s):  
A. N. Shevlyakov
Keyword(s):  
Left Regular ◽  
Free Left ◽  
Left Regular Bands

Extensions and covers for semigroups whose idempotents form a left regular band

2010 ◽  
Vol 81 (1) ◽  
pp. 51-70 ◽  
Author(s):  
Mário J. J. Branco ◽  
Gracinda M. S. Gomes ◽  
Victoria Gould
Keyword(s):  
Regular Band ◽  
Left Regular ◽  
Left Regular Band

Wreath Product Structure of Left C-rpp Semigroups

2008 ◽  
Vol 15 (01) ◽  
pp. 101-108 ◽  
Author(s):  
Xiaojiang Guo ◽  
Ming Zhao ◽  
K. P. Shum
Keyword(s):  
Wreath Product ◽  
Product Structure ◽  
Inverse Semigroups ◽  
Regular Band ◽  
Clifford Semigroups ◽  
Left Regular ◽  
Left Regular Band

The concept of wreath product of semigroups was first introduced by Neumann in 1960, and later on, this concept was used by Preston to investigate the structure of some inverse semigroups. In this paper, we modify the wreath product given by Neumann and Preston to study the structure of some generalized Clifford semigroups. In particular, we prove that a semigroup is a left C-rpp semigroup if and only if it is the wreath product of a left regular band and a C-rpp semigroup. Our result provides a new insight to the structure of left C-rpp semigroups.

scholarly journals THE QUIVER OF THE SEMIGROUP ALGEBRA OF A LEFT REGULAR BAND

2007 ◽  
Vol 17 (08) ◽  
pp. 1593-1610 ◽  
Author(s):  
FRANCO V. SALIOLA
Keyword(s):  
Random Walks ◽  
Hyperplane Arrangement ◽  
Semigroup Algebra ◽  
Module Structure ◽  
Regular Band ◽  
Combinatorial Objects ◽  
Left Regular ◽  
The Face ◽  
Left Regular Band ◽  
Cartan Invariants

Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan invariants are computed. These are used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and to show that the semigroup algebra of the free left regular band is isomorphic to the path algebra of its quiver.

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