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

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

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].

In and out of the Deepest Depression

Rollini joins Richard Himber' Ritz-Carlton hotel band and does some recording work under his own name, but he wants more stability and decides to start a club of his own. This became Adrian's Tap Room where Fats Waller was the piano player/entertainer. Musicians would gather there and often jam together and with the regular band. Also Rollini crossed paths again with Ed Kirkeby who, as an agent for Victor, got him several record dates for that label.With some dates with ARC a.o. with Dick McDonough, Rollini's dark Depression years ended.

Defect Detection on Patterned Fabrics Using Distance Matching Function and Regular Band

In this paper, regular band is presented to detect defects on patterned fabrics. Patterned fabrics are firstly disposed by fabric average to form object images mixed with positive and negative pixels in this proposed method. Distance matching function is computed to determine the periodic distance of patterned fabrics. The obtained periodic distance would be the length and width of regular band on row and column. Two features are calculated with regular band. The threshold of defect segmentation is extracted from the training step. Two features of regular band negotiating the threshold are considered as a fabric defect. Regular band method in this paper can avoid patterned interference and obtain perfect results in defect detection of patterned fabrics. Various defects on patterned fabrics can be inspected, and speedy defect detection could prove the performance ability in practice.

$\widetilde{\cal H}$-Supercryptogroups Having Regular Band Congruence

We consider a generalized superabundant semigroup within the class of semiabundant semigroups, called a supercryptogroup since it is an analogy of a cryptogroup in the class of regular semigroups. We prove that a semigroup S is an [Formula: see text]-regular supercryptogroup if and only if S can be expressed as a refined semilattice of completely [Formula: see text]-simple semigroups. Some results on regular cryptogroups are extended to [Formula: see text]-regular supercryptogroups. Some results on superabundant semigroups are also generalized.

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

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.