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.

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 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.

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.

Lawrence Polytopes

1990 ◽  
Vol 42 (1) ◽  
pp. 62-79 ◽  
Author(s):  
Margaret Bayer ◽  
Bernd Sturmfels
Keyword(s):  
Oriented Matroids ◽  
Oriented Matroid ◽  
Face Lattice ◽  
The Face

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a discussion of this construction and both of these applications.

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 Diameters of Cocircuit Graphs of Oriented Matroids: An Update

10.37236/9653 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ilan Adler ◽  
Jesús A. De Loera ◽  
Steven Klee ◽  
Zhenyang Zhang
Keyword(s):  
Directed Graphs ◽  
Hyperplane Arrangements ◽  
Oriented Matroids ◽  
Low Rank ◽  
Oriented Matroid ◽  
Computer Search ◽  
Linear Programs ◽  
Large Computer ◽  
Positive Side ◽  
Point Configurations

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key  role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof).  For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false. 

scholarly journals Strong bands of groups of left quotients

1996 ◽  
Vol 38 (2) ◽  
pp. 237-242
Author(s):  
Miroslav Ćirić ◽  
Stojan Bogdanović
Keyword(s):  
General Result ◽  
Group Inverse ◽  
Regular Band ◽  
Principal Problem ◽  
Strong Band ◽  
Left Regular ◽  
Left Regular Band ◽  
Left Order

An interesting concept of semigroups (and also rings) of (left) quotients, based on the notion of group inverse in a semigroup, was developed by J. B. Fountain, V. Gould and M. Petrich, in a series of papers (see [5]-[12]). Among the most interesting are semigroups having a semigroup of (left) quotients that is a union of groups. Such semigroups have been widely studied. Recall from [3] that a semigroup has a group of left quotients if and only if it is right reversible and cancellative. A more general result was obtained by V. Gould [10]. She proved that a semigroup has a semilattice of groups as its semigroup of left quotients if and only if it is a semilattice of right reversible, cancellative semigroups. This result has been since generalized by A. El-Qallali [4]. He proved that a semigroup has a left regular band of groups as its semigroup of left quotients if and only if it is a left regular band of right reversible, cancellative semigroups. Moreover, he proved that such semigroups can be also characterised as punched spined products of a left regular band and a semilattice of right reversible, cancellative semigroups. If we consider the proofs of their theorems, we will observe that the principal problem treated there can be formulated in the following way: Given a semigroup S that is a band B of right reversible, cancellative semigroups Si, i ε B, to each Si, we can associate its group of left quotients Gi. When is it possible to define a multiplication of such that Q becomes a semigroup having S as its left order, and especially, that Q becomes a band B of groups Gi, i E B?Applying the methods developed in [1] (see also [2]), in the present paper we show how this problem can be solved for Qto become a strong band of groups (that is in fact a band of groups whose idempotents form a subsemigroup, by [16, Theorem 2]. Moreover, we show how Gould's and El-Quallali's constructions of semigroups of left quotients of a semilattice and a left regular band of right reversible, cancellative semigroups, can be simplified.

Sign in / Sign up

Export Citation Format

Share Document