Stuart Margolis
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Master List of Examples in Complexity Theory of Finite Semigroup Theory Open
This document gives a list of finite semigroups that are interesting from the point of view of Krohn-Rhodes complexity theory. The list will be expanded and updates as "time goes by".
Complexity of Finite Semigroups: History and Decidability Open
In recent papers, Margolis, Rhodes and Schilling proved that the complexity of a finite semigroup is computable. This solved a problem that had been open for more than 50 years. The purpose of this paper is to survey the basic results of K…
Aperiodic Flows on Finite Semigroups: Foundations and First Examples Open
The theory of flows was used as a crucial tool in the recent proof by Margolis, Rhodes and Schilling that Krohn-Rhodes complexity is decidable. In this paper we begin a systematic study of aperiodic flows. We give the foundations of the th…
Decidability of Krohn-Rhodes complexity for all finite semigroups and automata Open
The Krohn-Rhodes Theorem proves that a finite semigroup divides a wreath product of groups and aperiodic semigroups. Krohn-Rhodes complexity equals the minimal number of groups that are needed. Determining an algorithm to compute complexit…
View article: Topics in Boolean Representable Simplicial Complexes
Topics in Boolean Representable Simplicial Complexes Open
We study a number of topics in the theory of Boolean Representable Simplicial Complexes (BRSC). These include various operators on BRSC. We look at shellability in higher dimensions and propose a number of new conjectures.
On the minimal faithful degree of Rhodes semisimple semigroups Open
In this paper we compute the minimum degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all …
Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata Open
When decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups, complexity measures the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for alm…
Krohn-Rhodes complexity 1 decidable implies that complexity $n \geqslant 0$ is decidable Open
When decomposing a finite semigroup into a wreath product of groups and aperiodic semigroups, complexity measures the minimal number of groups that are needed. Determining an algorithm to compute complexity has been an open problem for alm…
Degree 2 transformation semigroups as continuous maps on graphs: Foundations and structure Open
We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action gi…
Ehresmann Semigroups Whose Categories are EI and Their Representation\n Theory : Extended Version Open
We study simple and projective modules of a certain class of Ehresmann\nsemigroups, a well-studied generalization of inverse semigroups. Let $S$ be a\nfinite right (left) restriction Ehresmann semigroup whose corresponding\nEhresmann categ…
Ehresmann Semigroups Whose Categories are EI and Their Representation Theory : Extended Version Open
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category…
View article: On the Wilson monoid of a pairwise balanced design
On the Wilson monoid of a pairwise balanced design Open
We give a new perspective of the relationship between simple matroids of rank 3 and pairwise balanced designs, connecting Wilson’s theorems and tools with the theory of truncated boolean representable simplicial complexes. We also introduc…
View article: Truncated boolean representable simplicial complexes
Truncated boolean representable simplicial complexes Open
We extend, in significant ways, the theory of truncated boolean representable simplicial complexes introduced in 2015. This theory, which includes all matroids, represents the largest class of finite simplicial complexes for which combinat…
Ehressman Semigroups Whose Categories are EI and Their Representation Theory Open
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category i…
The lattice of flats of a boolean representable simplicial complex Open
It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic latti…
The algebra of the Catalan monoid as an incidence algebra: A simple proof Open
We give a direct straightforward proof that there is an isomorphism between the algebra of the Catalan monoid C_n that is, the monoid of all order-preserving, weakly increasing self-maps f of [n] = {1,...,n}, over any commutative ring with…
Projective indecomposable modules and quivers for monoid algebras Open
We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent genera…
On the subsemigroup complex of an aperiodic Brandt semigroup Open
We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated throug…
On the topology of a boolean representable simplicial complex Open
It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension [Formula: see text…
On the topology of a boolean representable simplicial complex Open
It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension >= 2. In the case …